Coarse-grained models of polymer and biomolecular systems have enabled the computational study of cooperative phenomena, e.g., self-assembly, by lumping multiple atomistic degrees of freedom along the backbone of a polymer, lipid, or DNA molecule into one effective coarse-grained interaction center. Such a coarse-graining strategy leaves the number of molecules unaltered. In order to treat the surrounding solvent or counterions on the same coarse-grained level of description, one can also stochastically group several of those small molecules into an effective, coarse-grained solvent bead or “fluid element.” Such a procedure reduces the number of molecules, and we discuss how to compensate the concomitant loss of translational entropy by density-dependent interactions in spatially inhomogeneous systems.

The development of coarse-grained models has a longstanding tradition in polymer science. Early renormalization group approaches established the universality of macromolecular conformations in solutions and melts and provided insights into the scaling behavior as a function of the chain length.1,2 Subsequently, attention focused on systematic coarse-graining strategies, where a small number of atomistic repeating units along the backbone of a macromolecule,3–5 a lipid,6–11 or a DNA molecule12–14 are lumped together into an effective, coarse-grained interaction center—fine-to-coarse map. This coarse-graining strategy decimates the number of degrees of freedom (DoFs), and the coarse-grained particles interact via soft potentials. Both effects permit the investigation of significantly longer time and larger length scales and thereby enable the systematic investigation of cooperative phenomena in soft matter.9,15

Coarse-graining gives rise to interactions between the coarse-grained particles that depend on the thermodynamic state and may include multi-body or density-dependent terms.15–20 Significant practical progress has been achieved in deriving the interaction free energies between the coarse-grained particles from an underlying atomistic model by iterative Boltzmann inversion (IBI),19,21,22 inverse Monte Carlo (IMC) schemes,23,24 or force matching,18,25 and a unifying framework, based on the relative entropy26,27 of the atomistic and coarse-grained description, has been devised. More recently, data-driven formulations, using a probabilistic coarse-to-fine map, have been explored which do not require an explicit fine-to-coarse map.28 

Work on the foundation of Dissipative Particle Dynamics (DPD) has also considered the dynamics of coarse-grained particles that capture the behavior of a simple fluid of non-bonded microscopic constituents.29–32 The DoFs that have been integrated out give rise to friction and noise in the coarse-grained model, and the energy and entropy of these decimated DoFs can be captured by endowing each coarse-grained particle with an additional, internal-energy DoF, e, and a concomitant internal entropy, s(e). Locally conserving the total energy comprised of the kinetic and potential energy of the explicit DoF and the internal energy, these eDPD,33,34 eMC,35 or related36 simulation schemes capture the specific heat and thermal transport properties. The exchange of internal energies between neighboring coarse-grained particles during an eMC move is controlled by their internal entropy, s(e), and the internal DoF, e, also couples to the equation of motion of the explicit DoFs, i.e., the positions and momenta of the coarse-grained particles. Alternatively, the internal DoF may characterize multiple metastable states (e.g., discrete molecular conformations or chemical states) “within” a coarse-grained particle.37 

Whereas many coarse-graining schemes of polymer and biomolecular systems group the microscopic DoF along the same molecule into a coarse-grained particle, it may also be advantageous to treat the surrounding solvent molecules on the same level of coarse-graining by lumping multiple solvent molecules into one coarse-grained particle.38 The most drastic approach is an implicit solvent model, where the complete elimination of the solvent DoFs results in density-dependent effective interactions between the remaining particles.10,39,40 Another example is the MARTINI8 description of water, where 4 water molecules are grouped together into a single, coarse-grained particle41,42 or the recently devised cDPD algorithm43 for electrokinetic phenomena, which represents a fluctuating number of ions by a coarse-grained particle.

Coarse-graining approaches that map multiple, indistinguishable, microscopic molecules into a coarse-grained particle do not conserve the number of molecules but often do not consider the loss of translational entropy that goes along with the decimation of DoFs. In a spatially homogeneous system, this omission can be compensated by state-dependent, coarse-grained interactions, in particular one-body terms that are independent of the microscopic configuration.17,44 The lack of transferability, however, becomes even more important in spatially inhomogeneous systems.17,19,45 This issue can be illustrated by the simple example of an ideal gas of particles with mass m in a gravitational field. In thermodynamic equilibrium, the density distribution, ρ(z), of the microscopic particles as a function of altitude, z, follows the barometric formula, ρ(z) ∼ exp(−mgz/kBT) with kBT being the thermal energy scale. This distribution arises from a competition between the energy in the gravitational field, which favors the accumulation of particles at low altitudes, and the translational entropy that encourages the particles to explore space uniformly. Obviously, we cannot capture this behavior in a simple, coarse-grained model where each coarse-grained particle is comprised of q¯=2 microscopic particles and has the mass q¯m. While such a coarse-grained model faithfully captures the energy in the gravitational field, it fails to account for the loss of translational entropy due to the reduction of the number of indistinguishable particles. In spatially inhomogeneous polymeric and biomolecular systems, similar balances between the translational entropy and energy are ubiquitous, e.g., in the adsorp tion of particles onto surfaces or onto large biomolecules, or the partitioning of particles between coexisting phases.

In this manuscript, we study the statistical mechanics of decimating the number of indistinguishable, microscopic molecules and discuss how to account for the concomitant loss of translational entropy by density-dependent potentials. Each coarse-grained particle, β, is characterized by its explicit DoF (e.g., position, rβ) and, additionally, a fluctuating, discrete, internal DoF, qβ, representing the number of indistinguishable, microscopic molecules contained in it. The fine-to-coarse map is stochastic, i.e., one configuration of indistinguishable microscopic molecules may be represented by multiple configurations of coarse-grained particles. Explicit expressions are obtained using a lattice of subvolumes, and a generalization to an approximate off-lattice description is outlined.

In our manuscript we use simple algorithms—random particle displacements, Monte Carlo moves that transfer a microscopic particle from one coarse-grained particle to another, or Monte Carlo moves that swap the contents qi and qj between two coarse-grained particles, i and j—that suffice to appropriately sample configuration space. More sophisticated algorithms could be devised inspired by Monte Carlo algorithms for supramolecular systems,46–48 polydisperse systems,49,50 or the recently devised adaptive resolution approach for coupling atomistic water models with supramolecular ones.42 

Both lattice-based and off-lattice coarse-grained models are quantitatively validated by simulating two one-dimensional model systems—(a) repulsive particles in a gravitational field and (b) cylindrically symmetric distribution of counterions around an oppositely charged wire. These systems were chosen to highlight the role of translational entropy: The pairwise interactions do not pose a challenge for coarse-graining, the systems exhibit mean-field-like behavior, and explicit, analytical predictions for their equilibrium density distributions are available for a quantitative comparison.

The outline of the manuscript is as follows: In Sec. II, we derive classes of coarse-grained models by stochastically mapping multiple, indistinguishable molecules into a coarse-grained particle. Depending on the coarse-graining scheme, the number, qβ, of microscopic particles that are comprised in the coarse-grained particle, β—the degree of coarse-graining—may be constant or fluctuate, and we also discuss schemes that allow us to control the spatial variation of qβ in order, e.g., to obtain a less coarse-grained description in the vicinity of a surface and a more aggressive coarse-graining farther away. At the end of Sec. II, we indicate an approximate scheme that allows us to avoid the lattice of non-overlapping subvolumes, which is employed to quantify the translational entropy. In Sec. III, we apply the lattice-based and off-lattice coarse-grained models to the two test systems. Comparing the results with the exact analytic predictions we find excellent agreement. Sec. IV details our conclusions and presents a brief outlook on potential future applications.

We consider a system of N microscopic particles—gas molecules, solvents, or counterions—with coordinates, {Rα}, in a fixed volume, V, at constant temperature, T. A typical system configuration is depicted in the left panel of Fig. 1. The canonical partition function of the original system of interacting, microscopic particles is given by

Z=1λ3NN!α=1NdRαexpH({Rα})kBT,
(1)

where H denotes the total interaction energy as a function of the microscopic particle coordinates, which is comprised of the potential energy in an external field and pairwise interactions between the microscopic particles. We have chosen to normalize the conformational partition function by a generic length λ that sets the unit of length. For the particles in the gravitational field, this length scale is the thermal de Broglie wavelength, whereas we identify it with the Bjerrum length in the case of counterions around a charged wire.

FIG. 1.

Illustration of a configuration of N = 17 indistinguishable, microscopic particles in a volume V that is divided into M = 4 subvolumes. The left panel depicts the configuration, {Reα}, of the original model. The middle panels present two alternate representations via coarse-grained particles, rβ,ηβ. The different symbols and colors indicate the corresponding, coarse-grained particles, n = 9 (top) and n = 8 (bottom). The right panels show the corresponding configurations of the coarse-grained model, {rβ, qβ}.

FIG. 1.

Illustration of a configuration of N = 17 indistinguishable, microscopic particles in a volume V that is divided into M = 4 subvolumes. The left panel depicts the configuration, {Reα}, of the original model. The middle panels present two alternate representations via coarse-grained particles, rβ,ηβ. The different symbols and colors indicate the corresponding, coarse-grained particles, n = 9 (top) and n = 8 (bottom). The right panels show the corresponding configurations of the coarse-grained model, {rβ, qβ}.

Close modal

We define a dimensionless density, ρ^(r), of microscopic particles by

ρ^(r)=λ3α=1Nδ(rRα),
(2)

where the “hat” indicates that this quantity depends on the configuration of microscopic particles, {Rα}. The external and pairwise interactions can be expressed as a functional of the microscopic density, H({Rα})=H[ρ^], a relation that is exploited to devise top-down, coarse-grained models.51 

For bookkeeping purposes, let us subdivide the total volume, V, of the system into M non-overlapping subvolumes, Vi, and let Ni denote the number of microscopic particles in the ith volume element. i=1MNi=N with Ni=Vidrα=1Nδ(rRα), and i=1MVi=V. The position of the αith microscopic particle in subvolume Vi is denoted by Riαi. With these definitions, we can rewrite the partition function in the form

Z=1λ3NN!{Ni}N!i=1MNi!i=1MViαi=1NidRiαieH({Rα})kBT=1λ3N{Ni}i=1M1Ni!Viαi=1NidRiαieH({Rα})kBT,
(3)

where the multinomial N!/(i=1MNi!) quantifies the number of possibilities to distribute N identical, microscopic particles so that the ith subvolume contains Ni microscopic particles. The integral over the microscopic coordinate, Riαi, is confined to the subvolume, Vi. Here and in the following, it is understood that the sum over {Ni} is restricted to combinations that fulfill the condition

i=1MNi=N.
(4)

We group the Ni microscopic particles in subvolume Vi into ni coarse-grained particles as illustrated in the middle panel of Fig. 1. In contrast to Voronoi fluid particles,30,31 the coarse-grained particles may overlap, i.e., typically, we assume that a subvolume contains multiple coarse-grained particles. Note that such a fine-to-coarse map is not deterministic but stochastic, i.e., there are different possible ways to represent a microscopic configuration of indistinguishable particles by coarse-grained particles, and we assign an a priori probability (see below) to the different alternate representations.

In the following, we assume that the total number of coarse-grained particles, n=i=1Mni, is constant (i.e., the canonical ensemble of the coarse-grained model). The coarse-grained particle with index i in Vi is comprised of qiβi microscopic particles that are all located inside the same subvolume Vi. Thus the subvolume defines the “size” of a coarse-grained particle, i.e., the spatial region, from which microscopic particles can be recruited into a coarse-grained one.52qiβi=αiβi=1qiβi1 where αiβi=1,,qiβi enumerates the microscopic particles that are grouped into the coarse-grained particles, i. We select one, αiβi*, of the qiβi microscopic particles to define the explicit position, riβi, of the coarse-grained particle, whereas the qiβi1 other coordinate vectors define the internal positional degrees of freedom (DoFs), {η}iβi, of the coarse-grained particle, i,53 

riβiRαiβi*andηαiβi=Rαiβi   for   αiβiαiβi*.
(5)

This choice of the explicit positional DoF, rβ, as primus inter pares has the advantage that if all coarse-grained particles are comprised of the same number of microscopic particles, qβ=q¯ see Sec. II D 1, the average densities of microscopic and coarse-grained particles are simply related by ρ^(r)=q¯ρ^cg(r). The variables riβi,{η}iβi with i = 1, , M and βi = 1, , ni are a complete, alternative description of the microstate of the system of microscopic particles, and the Jacobian determinate of this mapping from {Rα} to {riβi,{η}iβi} is unity.

Bock and co-workers38 also considered the coarse-graining of two (solvent) particles into one effective interaction center in a homogeneous fluid. They lumped any two particles together into a coarse-grained particle and characterized the resulting coarse-grained particle by the average position (midpoint) of the two constituent, microscopic particles. In the thermodynamic limit, the typical distance between the two constituent particles is much larger than the finite range of interactions or the fluid-like packing effects in the liquid. Using a rigorous approach, they explicitly demonstrated that the coarse-grained particle contributes to the Hamiltonian only via a constant, chemical potential term. In turn, we consider a spatially inhomogeneous system, require that the two particles which are lumped together into a coarse-grained particle are located within the same subvolume, and characterize the position of the coarse-grained segment by the position of one of the constituent microscopic particles.

As illustrated by the two middle panels of Fig. 1, the number, Ni, of microscopic particles in the ith subvolume does not uniquely determine the number, ni, of coarse-grained particles in Vi. Instead, for a given configuration of microscopic particles, the number, ni, of coarse-grained particles and the numbers, qiβi, of microscopic particles contained in the coarse-grained particle, i, fluctuate, subjected to the constraints that (i) the total number of coarse-grained particles in the entire system amounts to n and (ii) the total number of microscopic particles in subvolume Vi is Ni

βi=1niqiβi=Nifor all i=1,,M.
(6)

Thus configurations of the microscopic model can be represented in the form {rβ, {η}β}, where {η}β collectively denotes the qβ − 1 internal, positional DoFs of the coarse-grained particle, β. In particular, this representability condition requires that there is at least one coarse-grained particle in each subvolume that is populated by microscopic particles, i.e., ni > 0 if Ni > 0. The number, qiβi, of microscopic particles that are lumped into the coarse-grained particle, i, can adopt non-negative, integer values including 0, corresponding to an “empty” coarse-grained particle.

The mapping procedure from the microscopic system, {Rα}, to an alternate representation, {riβi,{η}iβi}, as illustrated in Fig. 1, specifies the grouping of the microscopic particles into the coarse-grained ones in each of the subvolumes, Vi. To a given distribution, {Ni}, of microscopic particles in the different subvolumes, we assign a distribution {ni} of coarse-grained particles with an a priori probability, PnM{Ni}(0)({ni}). For given distributions of microscopic and coarse-grained particles, {Ni} and {ni}, in subvolume Vi, in turn, a particular set of numbers of internal, positional DoFs {qiβi}=qi1,,qini of the coarse-grained particles is assigned the a priori probability, pNini(0)({qiβi}). Note that pNini(0)({qiβi}) does not account for the number of ways the Ni microscopic particles can be grouped into ni coarse-grained ones [see Eq. (10)]. Both a priori distributions are normalized,

{ni}PnM{Ni}(0)({ni})=1  {Ni},
(7)
{qiβi}pNini(0)({qiβi})=1  ni and Ni,
(8)

where, here and in the following, the first sum is restricted to fulfill the condition, i=1Mni=n, and the second one obeys the constraint, Eq. (6). Apart from these normalizations and the representability condition, Eq. (6), the a priori distributions are arbitrary, and we shall discuss different choices in Sec. II D.

Each configuration, {rβ, {η}β}, corresponds to a valid configuration of N^=β=1nqβ indistinguishable, microscopic particles according to Eq. (3). Here and in the following, the “hat” indicates that the quantity is computed from the coarse-grained configuration. The number of microscopic particles, N^i, in subvolume Vi is computed from the position of the coarse-grained particles, {rβ}, and their number of internal DoFs, {qβ}, according to

N^iβi=1n^iqiβi=β=1nqβδi,i(rβ),
(9)

where i(rβ) denotes the index of the subvolume in which the coarse-grained particle, β, is located. Conversely, however, it is not automatically ensured that, for an arbitrary choice of a priori distributions and subvolumes, the configurations {rβ, {η}β} can represent all typical configurations of the N^ microscopic particles {Rα} because all qβ microscopic particles associated with the coarse-grained particle, β, must be located in subvolume, i(rβ).54 For example, consider the extreme case that all coarse-grained particles are comprised of the same, large number of microscopic particles, qβ=q¯1, see Sec. II D 1. In this extreme case, each subvolume contains a multiple of q¯ microscopic particles, which would not be a typical representation of an ideal gas of microscopic particles if the average density, N/V, is not much larger than q¯/Vi, i.e., the typical microscopic values of Ni cannot be represented by qiβi according to Eq. (6). In the following, we assume that the a priori distributions and subvolumes have been appropriately chosen so that the typical configurations of the microscopic model can be represented by the alternate representation—representability condition on the coarse-grained DoFs.

Defining an alternate representation, {riβi,{η}iβi}, of the microscopic configuration in subvolume Vi, we group the Ni indistinguishable, microscopic particles into ni coarse-grained particles comprised of qiβi microscopic particles. The number of possible ways of grouping is given by the multinomial Ni!/βi=1niqiβi!. Furthermore, for each coarse-grained particle, the explicit position, riβi, has to be chosen from the qiβi possibilities and the remaining qiβi1 internal positions have to be indexed, yielding qiβi! possibilities. Thus, a given configuration of Ni microscopic particles in Vi can be assigned in Ni! distinct ways. Since the microscopic particles are indistinguishable, the different ways of mapping must not contribute to the entropy. Thus the configurational integral over the Ni indistinguishable, microscopic particles in Vi corresponds to the configurational integral over the ni coarse-grained particles in volume Vi and the configurational integral over the internal, positional DoFs,

1Ni!Viαi=1NidRαi=βi=1niqiβi!Ni!inversenumberofgroupingsViβi=1nidriβid{η}iβiqiβi!indistinguishable,coarsegrainedparticlesandinternalDoFs=1Ni!Viβi=1nidriβid{η}iβi.
(10)

Using Eq. (3), we write the partition function in the form

Z={Ni}{ni}PnM{Ni}(0)({ni})i=1M{qiβi}pNini(0)({qiβi})1Ni!×Viβi=1nidriβid{η}iβiλ3qiβieH({riβi,{η}iβi})kBT.
(11)

In the course of coarse-graining, the internal, positional DoFs, {η}β, are integrated out so that, finally, a coarse-grained particle is only characterized by its position, rβ, and the number of positional DoFs, qβ. A typical configuration of the coarse-grained model, {rβ, qβ} is presented in the right panel of Fig. 1. Splitting the integration over the explicit and internal, positional DoF and re-summing the former, we obtain the final expression for the partition function of the coarse-grained model,

Z={Ni}{ni}PnM{Ni}(0)({ni})i=1M{qiβi}pNini(0)({qiβi})1Ni!Viβi=1nidriβid{η}iβiλ3qiβieH({riβi,{η}iβi})kBT
(12)
=1n!{ni}n!i=1Mni!i=1MViβi=1nidriβiλ3{Ni}PnM{Ni}(0)({ni})i=1M{qiβi}pNini(0)({qiβi})ni!Ni!Viβi=1nid{η}iβiλ3(qiβi1)eHkBT
(13)
=1n!{qβ}β=1nVdrβλ3β=1nVi(rβ)d{η}βVi(rβ)qβ1eH({rβ,{η}β})kBTelnPnM{N^i}(0)({n^i})i=1MlnpN^in^i(0)({qiβi})+lnN^i!λ3ViN^ilnn^i!λ3Vin^i,
(14)
Z=1n!{qβ}β=1nVdrβλ3eFcg({rβ,qβ})+F(0)TSQ+FidFidcgkBT,
(15)

where i(rβ) denotes the index of the subvolume that contains the coarse-grained particle β, and N^i and n^i are the number of microscopic and coarse-grained particles in subvolume Vi, calculated from the explicit coordinates, {rβ}, of the coarse-grained particles. Moreover, the sum of {qβ} is constrained by the condition β=1nqβ=N; see Eqs. (4) and (6). In  Appendix A, we explicitly demonstrate how to obtain the partition function of an ideal gas for H=0 from Eq. (14).

The DPD community has made some efforts as well to discuss the problem of lumping non-bonded particles. To the best of our knowledge, however, the additional free-energy terms in Eq. (15) have not been considered but, instead, the focus has been on the general form and thermodynamic consistency of the equations of motion.29–32 

The different terms in the Boltzmann factor of Eq. (15) have a simple interpretation: The first term, Fcg, is the effective excess free energy that is obtained by integrating out the qβ − 1 internal, positional DoFs of the coarse-grained particles at fixed {rβ, qβ}, i.e.,

eFcg({rβ,qβ})kBTβ=1nVi(rβ)d{η}βVi(rβ)qβ1eH({rβ,{η}β})kBT.
(16)

Note that the internal, positional DoFs are constraint to remain within the subvolume of the associated coarse-grained particle,55 i.e., Fcg describes the effective interaction between clusters of microscopic particles of “size” Vi.56 The derivation of these effective interactions between the coarse-grained beads is an important challenge in its own right that has attracted abiding interest.17–19,21–23,25–27,57 In general, it contains density-dependent or multibody interaction. The computation of Fcg({rβ,qβ}) is not the focus of the present manuscript, and we have deliberately chosen applications, where an accurate approximation for the effective interactions can be obtained rather straightforwardly (see Sec. II E and  Appendix C). For instance, for an ideal gas of microscopic particles with mass m in a gravitational field g and a fine discretization (see Sec. II E), we obtain the intuitive result, Fcg({rβ,qβ})qβmgzβ.

The second term is associated with the possibilities of distributing the n coarse-grained particles into the M subvolumes so that each subvolume contains ni coarse-grained beads,

F(0)({rβ,qβ})kBT=lnPnM{N^i}(0)({n^i}).
(17)

Note that n^i and N^i can be computed from the coarse-grained configuration, {rβ, qβ}, according to Eq. (9).

The third term, SQ, is associated with the entropy of dividing the number of Ni microscopic particles in the subvolume Vi into the numbers qiβi (with βi = 1, , ni) of internal, positional DoFs of the ni coarse-grained particles. A particular realization, {qiβi}, occurs with probability pNini(0)({qiβi}). Thus, we obtain

SQ({rβ,qβ})kB=i=1MlnpNini(0)({qiβi}).
(18)

Specific choices for F(0) and SQ will be discussed in Sec. II D.

The fourth term, Fid, is the translational entropy of the microscopic particles or, equivalently, the free energy of the ideal gas of microscopic particles distributed over the M subvolumes,

Fid({rβ,qβ})kBT=i=1Mln1N^i!Viλ3N^ii=1MViλ3ρ^ilnρ^i1Vdrλ3ρ^(r)lnρ^(r)1,
(19)

where we have used Stirling’s formula, which is appropriate if each subvolume contains a sufficiently large number of microscopic particles, N^i1. The dimensionless densities of microscopic particles in the ith subvolume are given by

ρ^i=N^iλ3Vi=λ3Viβi=1n^iqiβi.
(20)

The “hat” again indicates that these quantities are calculated from the explicit configuration, {rβ, qβ}, of the coarse-grained model. Moreover, in the last expression, we have additionally approximated the subvolume-based density by its continuum limit, ρ^(r)N^i(r)λ3Vi(r), which is appropriate in the fine-discretization limit, see Sec. II E. Here i(r) is the index of the subvolume, in which the position r is located. These two approximations are mutually compatible only if the instantaneous density of the microscopic particles is high and slowly varying in space.

The fifth term, Fidcg, corresponds to the translational entropy of the coarse-grained particles,

Fidcg({rβ,qβ})kBT=i=1Mln1n^i!Viλ3n^iVdrλ3ρ^cg(r)lnρ^cg(r)1,
(21)

with the coarse-grained density distribution ρ^cg(r)n^i(r)λ3Vi(r). This term has to be subtracted because the translational entropy of the explicit, positional DoFs is generated by the configurational integral over the coarse-grained positions, {rβ}, but does not contribute to the free energy of the original system of microscopic particles. Thus, the difference, FidFidcg, quantifies the translational entropy of the Nn internal, positional DoFs that have been integrated out.

This exact rewriting, Eq. (15), of the partition function of the original N-body system of microscopic particles in terms of n coarse-grained particles with explicit positions, rβ, and a fluctuating number, qβ, of internal, positional DoFs, {η}β, is the central result of this manuscript.

Since the a priori distributions, PnM{Ni}(0)({ni}) and pNini(0)({qiβi}), only have to be normalized and obey the constraint, Eq. (6), different choices are possible, corresponding to different coarse-graining schemes. In this section, we discuss different schemes that are compiled in Table I.

TABLE I.

Summary of 10 different coarse-grained models that are obtained by distributing the coarse-grained particles among the volume elements according to PnM{Ni}(0)({ni}) and by lumping the Ni microscopic particles in subvolume Vi into ni coarse-grained particles according to pNini(0)({qiβi}).

PnM{Ni}(0)({ni}) orpNini(0)({qiβi}) or
F(0) in Eq. (17)SQ in Eq. (18)
Constant degree of coarse-graining, qβ=q¯ Equation (22) Equation (23) 
Gas of coarse-grained particles Equation (31) Bimodal Eq. (47) or unconstraint Eq. (49) 
Coarse-grained particles with varying fugacity Equation (34) Bimodal Eq. (47) or unconstraint Eq. (49) 
Uniform fluid elements, ni = 1 and qβ=Ni(rβ) Equation (39) Equation (40) 
Gas of fluid elements Equation (42) Bimodal Eq. (47) or unconstraint Eq. (49) 
Biased fluid elements with varying fugacity Equation (43) Bimodal Eq. (47) or unconstraint Eq. (49) 
PnM{Ni}(0)({ni}) orpNini(0)({qiβi}) or
F(0) in Eq. (17)SQ in Eq. (18)
Constant degree of coarse-graining, qβ=q¯ Equation (22) Equation (23) 
Gas of coarse-grained particles Equation (31) Bimodal Eq. (47) or unconstraint Eq. (49) 
Coarse-grained particles with varying fugacity Equation (34) Bimodal Eq. (47) or unconstraint Eq. (49) 
Uniform fluid elements, ni = 1 and qβ=Ni(rβ) Equation (39) Equation (40) 
Gas of fluid elements Equation (42) Bimodal Eq. (47) or unconstraint Eq. (49) 
Biased fluid elements with varying fugacity Equation (43) Bimodal Eq. (47) or unconstraint Eq. (49) 

1. Constant and equal number, q¯, of microscopic particles per coarse-grained particle

In the simplest case “constant degree of coarse-graining”, each coarse-grained particle is comprised of a fixed number, q¯, of microscopic particles.58 This choice of {rβ, {η}β} restricts the corresponding microscopic configurations, {Rα}, of the original particle-based model to those where the number of microscopic particles in each subvolume, Vi, is a multiple of q¯. As discussed above, such a restriction becomes irrelevant in the limit that each subvolume typically contains a large number of microscopic particles, Niq¯, but it becomes problematic if the density is low or the subvolumes are small. We will illustrate these representability issues in Sec. III A.

Since q¯ is constant, the number of microscopic particles, Ni, in subvolume, Vi, dictates the number of coarse-grained particles in this subvolume, niNi/q¯, and the a priori distributions take the form

PnM{Ni}(0)({ni})=i=1Mδni,Ni/q¯,
(22)
pNini(0)({qiβi})=1   ifqiβi=q¯iβi,and   0   otherwise.
(23)

For constant q¯, the sum over {qβ} in Eq. (15) contains only one term, and its a priori probability is pN^in^i(0)({qiβi})=1. Therefore, the charge-distribution entropy vanishes, SQ=0, and we obtain

Z=1n!β=1nVdrβλ3×β=1nVi(rβ)d{η}βVi(rβ)qβ1eH({rβ,{η}β})kBT=exp[Fcg/kBT]eFidFidcgkBT
(24)
=1n!β=1nVdrβλ3β=1nVi(rβ)d{η}βVi(rβ)qβ1×eH({rβ,{η}β})kBTi=1Mln(q¯n^i)!λ3Viq¯n^ilnn^i!λ3Vin^i.
(25)

Thus, the additional term in the exponential merely compensates for the difference between the translational entropies of the microscopic and coarse-grained particles.

It is instructive to describe a Monte Carlo scheme that samples the microscopic configurations characterized by {rβ, {η}β} according to the partition function, Eq. (25). Consider the attempt to move a randomly chosen coarse-grained particle, β*, which is comprised of q¯ microscopic particles, from its old position, rβ*, to a new position, rβ*. The displacement vector, Δrrβ*rβ*, is chosen from a symmetric distribution, pΔr) = pΔ(−Δr). The old and new positions of the coarse-grained particle are located in the subvolumes with indices, iold=i(rβ*) and inew=i(rβ*), respectively. The new positions, {η}β*, of the q¯1 internal, positional DoFs of the coarse-grained particle, β*, are randomly chosen within the new subvolume, Vinew. The probability to propose the Monte Carlo trial consisting of a displacement vector, Δr, and a specific set of new internal, positional DoF is poldnewprop=pΔ(Δr)/Vinewq¯1. Such a move implies a change of the interaction energy, ΔH, and the statistical weight, peq, of the configuration changes according to

pneweqpoldeq=[ninew+1]!(q¯[ninew+1])![niold1]!(q¯[niold1])!ninew!(q¯ninew)!niold!(q¯niold)!eΔHkBT=1nioldq¯nioldq¯1ninewq¯ninewq¯eΔHkBT,
(26)

where ninew=ninew+1 is the occupancy of the subvolume, Vinew, after the move has been accepted. Thus, within our coarse-grained simulation, we will accept this trial movement of the coarse-grained particle with the Metropolis acceptance criterion,

poldnewacc=min1,pneweqpoldeqpnewoldproppoldnewprop=min1,1nioldVioldq¯1q¯nioldq¯1ninewVinewq¯1q¯ninewq¯×eΔHkBT.
(27)

Conceiving this Monte Carlo move as a cluster move, we present an alternative derivation of this acceptance criterion in  Appendix B.

2. Choices for PnM{Ni}(0)({ni})—Coarse-grained particles vs fluid elements

In all other cases that we discuss in this section, qβ can adopt multiple values and thus there is no one-to-one relation between the number of microscopic and coarse-grained particles, Ni and ni, i.e., a microscopic configuration, {Rα}, can be represented by multiple coarse-grained configurations, {rβ, qβ}. The mapping between the microscopic and coarse-grained configurations is controlled by the a priori distributions. One can combine different choices of distributing the coarse-grained particles onto the subvolumes, PnM{Ni}(0)({ni}), with different schemes of grouping the microscopic particles into coarse-grained ones, pNini(0)({qiβi}), as summarized in Table I. In this section, we discuss two different options for the former a priori distribution, PnM{Ni}(0)({ni}).

a. Coarse-grained particles.

In order to link the number of coarse-grained particles to the number of microscopic particles, we choose the a priori probability

PnM{Ni}(0)({ni})=1i=1MZi(Ni)nn!i=1Mni!i=1MZi(Ni)ni,
(28)

where Zi(Ni) is an arbitrary function of the local number, Ni, of microscopic particles. PnM{Ni}(0)({ni})=0 if i=1Mnin. This a priori distribution corresponds to an ideal gas with a spatially varying fugacity, Zi(Ni)/Vi. The corresponding free energy takes the form59 

F(0)({rβ,qβ})kBTlnPnM{Ni}(0)({ni})
(29)
=n^lni=1MZi(N^i)lnn^!+Fidcg({rβ,qβ})kBTi=1Mn^ilnZi(N^i)λ3Vi.
(30)

In the following, we discuss two options for Zi(Ni):

  • The fugacity of an ideal gas is proportional to its density, i.e., Zi(Ni)/Viρ^icg. Therefore, choosing Zi = Ni, we correlate the density of the coarse-grained particles with the density of the microscopic particles60 and obtain the simple expression

F(0)({rβ,qβ})kBT=lnn!Nn+Fidcg({rβ,qβ})kBTi=1Mn^ilnρ^i.
(31)

The first term is an irrelevant constant that does not depend on the configuration of coarse-grained particles,{rβ, qβ}, the second term cancels the correction for the translational entropy of the coarse-grained particles in Eq. (15), and the third term represents the coupling between the coarse-grained and microscopic particles. In this case, we expect that the density of coarse-grained particles follows the density of microscopic particles. Thus, the “local degree of coarse-graining” defined by

q¯iρ^iρ^icg=β=1nqβδi,i(rβ)β=1nδi,i(rβ)
(32)

is rather uniform.

  • (ii)

    In order to control the local degree of coarse-graining, q¯i, as a function of the spatial position (e.g., in order to obtain a fine-grained description of the solvent in the vicinity of a macromolecule or a surface, and a coarser description farther away), we set Zi(Ni) = ϵiNi. The parameter, ϵi, is large in regions where the number of internal, positional DoFs per coarse-grained particle is low and, conversely, a small ϵi gives rise to a low ratio of the density of coarse-grained particles and microscopic particles, i.e., the coarse-grained particles have a larger qβ. In this case, we obtain

F(0)({rβ,qβ})kBT=lnn!i=1MϵiNin+Fidcg({ni})kBTi=1Mnilnϵiρi
(33)
=lnn!β=1nϵi(rβ)qβn+FidcgkBTβ=1nlnϵi(rβ)ρ^i(rβ),
(34)

which reduces to Eq. (31) for ϵi ≡ 1 for all 1 ≤ iM. The last equation indicates that the extensive free energy, F(0), is comprised of additive contributions from each coarse-grained particle, β.

b. Fluid elements: A priori de-correlation between microscopic and coarse-grained particles.

In the previous case, Sec. II D 2 a, the a priori distribution generated a correlation between the distribution of coarse-grained particles, {Ni}, and that of the microscopic particles, {ni}. In this subsection, we consider the opposite limit, i.e., there is no a priori correlation between the distribution of coarse-grained and microscopic particles because PnM{Ni}(0)({ni}) does not depend on {Ni}. In this limit, a coarse-grained particle merely corresponds to a fluid element, i.e., the physical information is not related to the spatial density of coarse-grained particles (which is uniform on scales larger than Vi) but rather to the number of DoFs, qiβi, that it contains. In this sense, it can be conceived as a discretized volume element for the simulation of the free-energy functional.

In the limit of fine discretization (FiDis), H({rβ,{η}β})FiDisHFiDis({Ni}), where the interaction energy is specified by the number, Ni, of microscopic particles in a subvolume and the dependence on the positions of the microscopic particles within the subvolume can be ignored, we can provide a general result for arbitrary a priori distributions by rewriting Eq. (12) in the form

ZFiDis={Ni}{ni}PnM{Ni}(0)({ni})i=1M{qiβi}pNini(0)({qiβi})1Ni!Viβi=1nidriβid{η}iβiλ3qiβieHFiDis({Ni})kBT
(35)
={Ni}eF({Ni})kBT{ni}PnM{Ni}(0)({ni})i=1M{qiβi}pNini(0)({qiβi})=1  {ni}and{Ni}
(36)
={Ni}eF({Ni})kBT{ni}PnM{Ni}(0)({ni})
(37
with        
F({Ni})=kBTi=1Mln1Ni!Viλ3Ni+HFiDis=Fid({Ni})+HFiDis({Ni}).
(38)

Thus, the probability distributions of the microscopic and the coarse-grained particles decouple, provided that PnM{Ni}(0)({ni}) does not explicitly depend on {Ni}. The distribution of the microscopic particles is dictated by the free-energy functional, F({Ni}), that accounts for the translational entropy of the coarse-grained particles and their interactions (within the fine-discretization approximation). The distribution of the coarse-grained particles is dictated by the a priori distribution, PnM{Ni}(0)({ni}). A local increase of the density of microscopic particles does not result in an increase of the density of the coarse-grained particles (or fluid elements) but rather to a change of their properties (i.e., the number of internal, positional DoFs, qβ).

In the following, we discuss three special cases: (i) one fluid element per subvolume, Vi, (ii) an ideal gas of fluid elements, and (iii) a biased spatial distribution of fluid elements.

  • In order to establish the connection to the field-based simulation of the discretized free-energy functional, we choose the a priori distribution of the coarse-grained particles so that each non-overlapping subvolume, Vi, is occupied by exactly one coarse-grained particle, i.e., n = M and ni = 1. In this case, there is a bijective correspondence between coarse-grained particles and subvolumes, and the number of internal, positional DoFs of the coarse-grained particle is identical to the number of microscopic particles in that subvolume, qiβi=Ni.

The a priori distributions for these “uniform fluid elements” take the form

PnM{N^i}(0)({n^i})=i=1Mδn^i,1,
(39)
pN^in^i(0)({qiβi})=1   for   qi1=Ni    and   0,   otherwise.
(40)

Starting from Eq. (12), we obtain for the partition function

Z={Ni}i=1M1Ni!Vidri1d{η}i1λ3NieH({riβi,{η}iβi})kBT{Ni}ei=1MlnNi!λ3ViNiHFiDiskBT={Ni}eF[Ni]kBT,
(41)

where, for the last two expressions, we have assumed that the discretization of the total volume into the M subvolumes is fine (cf. Sec. II E). Thus, we re-cover the field-based simulation of the original system of microscopic particles.

  • (ii)

    Contrarily, we can simply distribute the n coarse-grained particles randomly among the subvolumes, Vi. This procedure corresponds to the a priori distribution

PnM{Ni}(0)({ni})=n![V/λ3]ni=1Mni!i=1MViλ3ni,
(42)

which is identical to Eq. (28) with Zi(Ni) ≡ Vi/λ3 for all i. This special choice de-correlates the a priori distribution of the coarse-grained particles and the microscopic particles, i.e., the coarse-grained particles constitute a fluctuating set of sample points for the discretized free-energy functional. The free energy that corresponds to this a priori distribution takes the form [cf. Eq. (30)]

F(0)({ni})kBT=ln1n!Vλ3n+Fidcg({ni})kBT.
(43)

The two contributions correspond to the negative of the free energy of the ideal gas of fluid elements, which does not depend on the microscopic configuration, and the translational entropy of a specific configuration {ni}, of coarse-grained particles, respectively. Again, the extensive free-energy can be expressed in terms of additive contributions of the individual, coarse-grained particles.

  • (iii)

    Alternatively, we can exploit the choice of the a priori probability, PnM{Ni}(0)({ni}), to bias the spatial density of fluid elements, ρicg, and thereby, indirectly, also the local degree of coarse-graining, q¯, according to Eq. (32). A simple a priori distribution that allows us to tailor the coarse-grained density is

PnM{Ni}(0)({ni})=1i=1MViλ3eμikBTnn!i=1Mni!×i=1MViλ3eμikBTni,
(44)

where the a priori chemical potential, μi, of the coarse-grained particles in the subvolume Vi can be chosen arbitrarily. This a priori distribution corresponds to Eq. (28) with the special choice Zi(Ni)VieμikBT/λ3 for all i, and the corresponding free energy takes the form

F(0)({ni})kBT=ln1n!i=1MViλ3eμikBTn+Fidcg({ni})kBTi=1MμinikBT=ln1n!i=1MViλ3eμikBTn+Fidcg({ni})kBTβ=1nμi(rβ)qβkBT.
(45)

Note that the first term is irrelevant for the simulation because it does not depend on the microscopic configuration, and the second and third terms are additive.

3. Choices for pNini(0)({qiβi})—Bimodal vs unconstrained distributions

Unless the degree of coarse-graining is fixed to a constant value, q¯, for all particles or the number of coarse-grained particles per subvolume is constant (“uniform fluid elements”), there are different strategies, pNini(0)({qiβi}), to distribute the Ni microscopic particles onto the ni coarse-grained ones. In this section, we discuss two different options for the a priori distribution, pNini(0)({qiβi}):

a. Bimodal distribution of the number of internal, positional DoFs per coarse-grained particle.

In the first case, there are two types of coarse-grained particles, characterized by either qβ = 1 or qβ = q (with q > 1). The former particles are identical to the microscopic particles of the original model, whereas the latter correspond to proper, coarse-grained particles that are comprised of q microscopic ones. The distribution of the coarse-grained particles is not completely determined by that of the microscopic particles; for a given number of microscopic particles, Ni, in the ith subvolume the number of coarse-grained particles can vary from Ni/q to Ni.61 Therefore one can combine this pNini(0)({qiβi}) with one of the a priori distributions discussed in Sec. II D 2.

Given Ni = ni1 + qniq and ni = ni1 + niq, the numbers, ni1=qniNiq1 and niq=Niniq1 of both types of coarse-grained particles in subvolume Vi are completely determined. This case corresponds to the a priori distribution

pNini(0)({qiβi})=ni1!niq!ni!if the sequence,{qiβi},contains ni1 ones and niqqs,and 0,otherwise,
(46)

and the entropy takes the form

SQ({rβ,qβ})kBi=1MlnpNini(0)({qiβi})=i=1Mlnn^i1!n^iq!n^i!=Fidcg1+FidcgqFidcgkBT,
(47)

where Fidcg1 and Fidcgq denote the additive translational entropies of the coarse-grained particles with qβ = 1 and qβ = q, respectively.

b. Unconstrained fluctuations of qβ.

Limiting the number of internal, positional DoFs, qβ, we constrain the microscopic configurations, {Rα}, that can be represented by {rβ, {η}β}, particularly if the number of coarse-grained particles in a subvolume becomes small. Instead, we randomly group the Ni microscopic particles in subvolume Vi into ni coarse-grained particles, i.e., the a priori distribution, pNini(0)({qiβi}), and the concomitant extensive entropy takes the form

pNini(0)({qiβi})=Ni!niNiβi=1niqiβi!,
(48)
SQ({rβ,qβ})kB=β=1nlnqβ!β=1nqβlnρ^i(rβ)cg+FidkBTβ=1nqβlnqβN^i(rβ)/n^i(rβ),
(49)

where, in the last expression, we have used Sterling’s formula to indicate that SQ is associated with the deviation of the number qβ, of positional DoFs of a particle at rβ from its average, Ni/ni=q¯i, in the subvolume, in which the particle is located.

If all subvolumes, Vi, are small compared to the length scale, over which the physical properties of the system, e.g., density or external field, vary in space—the limit of fine spatial discretization, FiDis—the spatial extent of coarse-grained particles is irrelevant, and they can be conceived as coarse-grained point particles. This FiDis approximation is particularly appropriate if the microscopic particles do not exhibit local, fluid-like packing effects, i.e., there are no harsh, short-range, repulsive interactions as it is, e.g., the case for point-like particles of an ideal gas or if the microscopic particles interact via a soft potential. Then, one can approximate the density of microscopic particles by

ρ^(r|{Rα})λ3α=1Nδ(rRα)
(50)
=representabilityρ^(r|{rβ,{η}β})λ3β=1|qβ>0nδ(rrβ)+αiβi=1qβ1δ(rηαiβi)  ,
(51)
ρ^(r|{Rα})ρ^PIP(r|{rβ,qβ})λ3β=1nqβδ(rrβ)λ3β=1nρ^i(rβ)ρ^i(rβ)cgδ(rrβ),
(52)

where we have explicitly indicated, from which DoFs the density of microscopic particles is computed. Approximating ρ^(r|{Rα}) by ρ^(r|{rβ,{η}β}) will be accurate if the typical configurations, {Rα}, of the microscopic model are well represented by {rβ, {η}β}. The relation, ρ^(r|{rβ,{η}β})=ρ^(r|{rβ,qβ}), relies on the definition of the explicit, coarse-grained DoF (primus inter pares), asserting that the distribution of the explicit, positional DoF, rβ, and that of the qβ − 1 implicit, positional DoFs, {η}β, are identical in the subvolume, Vi(rβ). Thus, the average density of the microscopic particles can be estimated by rescaling the average density of the coarse-grained particles by a factor q̃i=ρ^iρ^icg, i.e., ρ^(r|{rβ,{η}β})q̃i(r)ρ^cg(r). Equation (52) provides an appropriate estimate of the density of microscopic particles also on scales that are smaller than a subvolume if the instantaneous density does not significantly fluctuate.

If the number of coarse-grained particles in a subvolume is small, it will be more accurate to account for the qβ − 1 internal, positional DoFs, {η}β, by their (unknown) distribution inside the subvolume Vi(rβ),

ρ^(r|{Rα})ρ^w(r|{rβ,qβ})λ3β=1nqβw(r,rβ),
(53)

where w(r, rβ) denotes the distribution of microscopic particles inside a subvolume that could be derived from a coarse-graining procedure.

If we assume that the physically relevant quantities slowly vary on the scale of a subvolume—the proper approximation of fine discretization—we can approximate the unknown distribution, w(r, rβ), of microscopic particles inside the subvolume Vi(rβ) by an index function, w(r,rβ)ΘVi(rβ)(r)/Vi(rβ) and obtain

ρ^(r|{Rα})ρ^FiDis(r|{rβ,qβ})=ρ^i(r)withρ^i=N^iλ3Vi=βi=1n^iqiβiλ3Vi.
(54)

In the following, we do not explicitly mention how the microscopic density is computed and simply denote this quantity as ρ^(r) because the argument, {Rα}, {rβ, {η}β}, or {rβ, qβ}, is set by the context.

This sequence of approximations allows us to express the density of microscopic particles as a function of the coarse-grained variables—the positions, {rβ}, and the number of internal, positional DoFs, {qβ}, of the coarse-grained particles—only; the internal positional DoFs, {{η}β}, are not required. Since the interaction energy, H, can be computed from the density of microscopic particles, we can use this approximation to calculate the interaction energy from the coarse-grained variables, {rβ, qβ},62 

H({Rα})=H({rβ,{η}β})=H[ρ^(r)]FiDisHFiDis({ρ^i})HFiDis({rβ,qβ}).
(55)

Since neither the grid-based density, ρ^i(r), of microscopic particles nor HFiDis depend on the qβ − 1 internal, positional DoFs, {η}β, of the coarse-grained particles, we use this fine-discretization approximation to integrate out the internal, positional DoFs in the partition function, Eq. (15), i.e., Vi(rβ)d{η}βVi(rβ)qβ1=1, and obtain

ZFiDis=1n!{qβ}β=1nVdrβλ3eHFiDis({rβ,qβ})+F(0)TSQ+FidFidcgkBT.
(56)

An appropriate choice for the size, Vi, of a subvolume or the range of the weighting function can be illustrated by the following representability consideration: In the special case of an ideal gas with a low density of coarse-grained particles, nVi/Vn/M ≤ 1, the typical configurations of the coarse-grained model consist of isolated clusters of qβ microscopic particles. For qβN/n > 1, such a coarse-grained configuration is not a typical configuration of the ideal gas of microscopic particles, i.e., the configurations {rβ, {η}β} do not well represent the microscopic configurations, {Rα}. Thus, the choice of ViV/M is a compromise: (a) Vi should be large enough for the coarse-grained particles to well represent the typical configurations of the microscopic particles and (b) Vi should be small enough in order to facilitate the computation of the coarse-grained interactions, Fcg, e.g., by the fine-discretization approximation. While the representability issue is inherent to the coarse-graining strategy, the calculation of Fcg can be systematically improved.

The explicit reference to the lattice of subvolumes gives rise to several disadvantages that are typical for lattice-based models: The lattice breaks translational and rotational invariance and imparts a lattice anisotropy onto the model. Moreover, the absence of forces does (a) not permit molecular dynamics simulation that is popular in biomolecular simulations, (b) prevents the calculation of the pressure via the virial of forces, and (c) complicates a hydrodynamic description (flow, shear, etc.).

Technically, the division into subvolumes is important for defining the free energies, F(0) and SQ, associated with the a priori distributions, as well as the translational entropies, Fid and Fidcg, which appear in the Boltzmann factor of the partition function, Eq. (15). All these free-energy contributions can be computed from the knowledge of {N^i} and {n^i} on the lattice of subvolumes or, equivalently, they only depend on the explicit configuration of the coarse-grained model {rβ, qβ} via the dimensionless, local densities of microscopic and coarse-grained particles, ρ^iN^iλ3/Vi and ρ^icgn^iλ3/Vi.

Importantly, being extensive free energies, these contributions are additive in the number of coarse-grained particles. The expressions for F(0) in Eqs. (34) and (45) as well as the entropy SQ in Eqs. (47) and (49) are comprised of sums over the coarse-grained particles, and each addend involves the number of internal, positional DoF, qβ, or the local densities of microscopic or coarse-grained particles, ρ^i(rβ) and ρ^i(rβ)cg, in the subvolume i(rβ), in which the particle is located. The translational entropies of microscopic and coarse-grained particles can be re-written in the same form

FidkBT=i=1MlnN^i!λ3ViN^i=i=1MN^ilnN^i!N^i+lnλ3Vi
(57)
=β=1nqβlnN^i(rβ)!N^i(rβ)+lnλ3Vi(rβ)=β=1nqβln[ρ^i(rβ)Vi(rβ)/λ3]!ρ^i(rβ)Vi(rβ)/λ3lnVi(rβ)λ3,
(58)
FidcgkBT=β=1nln[ρ^i(rβ)cgVi(rβ)/λ3]!ρ^i(rβ)cgVi(rβ)/λ3lnVi(rβ)λ3.
(59)

Since the lattice of subvolumes is only a technical aid, we can approximately replace the lattice-based density by a weighted density around the location of a coarse-grained particle, i.e., we approximate the density of microscopic particles in the surrounding of the coarse-grained particle, β, by a weighted density, ρ^(rβ), similar to multibody DPD techniques,10,63–65 embedded-atom potentials,66 or systematic coarse-graining schemes,18,57

ρ^i(rβ)λ3Vi(rβ)β=1nqβδi(rβ),i(rβ)β=1nqβw(rβ,rβ)ρ^(rβ).
(60)

The spatial extent, ΔV(rβ), of the weighting function, w(r, rβ), characterizes the size of a coarse-grained particle, and it obeys the normalization condition

drλ3w(r,rβ)=1.
(61)

Similarly, we can define the weighted densities of coarse-grained particles without making reference to a lattice of subvolumes

ρ^i(rβ)cgλ3Vi(rβ)β=1nδi(rβ),i(rβ)β=1nw(rβ,rβ)ρ^cg(rβ),
(62)

and analogous expressions hold for ρ^cg1(rβ) and ρ^cgq(rβ).

In the same spirit, we approximate the integral over the qβ − 1 internal, positional DoFs, {η}β, of the coarse-grained particle, β, by

Vi(rβ)d{η}βVi(rβ)qβ1Vd{η}βλ3(qβ1)αβ=1qβ1w(ηαβrβ)
(63)

and the size of a subvolume, Vi(rβ), in which the coarse-grained particle β is located is represented by ΔV(rβ).

Thus, the final approximate expression for the off-lattice partition function is

Z1n!{qβ}β=1nVdrβλ3×β=1nVd{η}βλ3(qβ1)αβ=1qβ1w(ηαβrβ)eH({rβ,{η}β})kBTexpFcgkBT×eF(0)TSQ+FidFidcgkBT,
(64)

where the free-energy contributions, Eqs. (57)–(59), are expressed in terms of the local off-lattice densities, Eq. (60).

In the following, we test the lattice and off-lattice, coarse-grained models using computer simulation of two example systems. These test systems were selected because (a) they highlight the role of translational entropy, (b) the interaction between the coarse-grained particles, Fcg({rβ,qβ}), is easily obtained, and (c) analytic results are available for a quantitative comparison because the systems are mean-field-like.

1. Definition and prediction of the mean-field behavior

We consider N microscopic particles at positions, {Rα}, that are subjected to an external, gravitational field and interact via a pair potential, V(r). The potential energy is given by H({Rα})=α=1NmgZα+12α,α=1NV(RαRα), where Zα is the coordinate of the αth microscopic particle in the direction of the gravitational field, and m and g being the mass of a microscopic particle and gravitational acceleration, respectively.67 Using the definition of the microscopic density, Eq. (2), we can rewrite the interaction energy in the form

H[ρ^]=drλ3mgzρ^(r)+12drdrλ6ρ^(r)V(rr)ρ^(r).
(65)

Alternatively, the structure and thermodynamics of the system can be described by a free-energy functional, F, of the density, ρ(r), of microscopic particles so that the canonical partition function, Z, takes the form

ZD[ρ]eF[ρ]kBT,
(66)

where the functional integral sums over all density fields, ρ(r), that satisfies the normalization constraint

drλ3ρ(r)=N=ξL2(Z0z0)λ3,
(67)

where ξ denotes the average number density of microscopic particles, L = Lx = Ly are the two spatial extensions of the system perpendicular to the direction of the gravitational field, and Z0 and z0 are the upper and lower boundaries of the system, respectively. For the system of pairwise interacting particles in a gravitational field, the free-energy functional is given by

F[ρ]kBT=drλ3ρ(r)lnρ(r)1+mgλkBTzλρ(r)+12drλ3ρ(r)V(rr)kBTρ(r)
(68)
=drλ3ρ(r)lnρ(r)1+mgλkBTzλρ(r)+v2λ3ρ2(r),
(69)

where the first term quantifies the translational free energy, the second term describes the interaction with the external, gravitational field, and the third contribution arises from the interactions between the microscopic particles. In the last line, we assume that the range of the repulsive pair potential is smaller than the characteristic length scales of all other system properties and set V(rr)=kBTvδ(rr), where v characterizes the excluded volume of a particle, which can be defined via the 2nd-order virial coefficient, v=B22 with B2=2πdrr2eV(r)kBT1 Thus, the interactions are characterized by two dimensionless parameters, mgλ/kBT and v/λ.3 

Additionally, we only consider one-dimensional variations of the density along the direction, z, of the gravitational field, i.e., we replace the three-dimensional density, ρ^(r), of microscopic particles by its laterally averaged analog, ρ^(z),

ρ^(z)=λ3L2α=1Nδ(zZα),
(70)

i.e., the microscopic particles “live” only in one spatial dimension and are only characterized by their altitude, Zα. The averaging over the lateral directions has two benefits: (a) It imparts a mean-field-like behavior onto the system that allows for an exact prediction of the density profile and (b) the “size” of the coarse-grained particles is large so that they strongly overlap, i.e., each subvolume contains many coarse-grained particles. The latter property, however, significantly slows down the off-lattice approach because one coarse-grained particle interacts with all particles within its ΔV = ΔzL2.

For this one-dimensional system, the free-energy functional takes the form

F[ρ]kBT=Lλ2z0Z0dzλρ(z)lnρ(z)1+mgzkBTρ(z)+v2λ3ρ2(z).
(71)

This one-dimensional system exhibits mean-field behavior in the limit L, N but ξ → const because (λ/L)2 plays the role of a Ginzburg parameter, i.e., the functional integral over the one-dimensional density fields in Eq. (66) is dominated by the density field that minimizes F[ρ] under the constraint that the system contains N microscopic particles; see Eq. (67). Minimization yields

μkBT=lnρ(z)+mgλkBTzλ+vλ3ρ(z),
(72)

where the chemical potential, μ, appears as a Lagrange multiplier of the constraint, Eq. (67). For a system of non-interacting particles—v = 0, ideal gas—we obtain the barometric formula ρ(z)=exp[μmgz]/kBT with exp(μ/kBT)=mgλ3N/L2exp(mgz0/kBT)exp(mgZ0/kBT). In the general case, v > 0, the density profile, ρ(z), is obtained numerically.

2. Coarse-grained simulations

We discretize the system extension, z0zZ0, in M uniform subvolumes, i.e., slabs of thickness, Δz=Z0z0M, and area, L2. The characteristic length scale of the system is set by the variations of the one-dimensional microscopic density profile, ρ^iρ(zi) with i = 1, …, M, where zi = z0 + [i − 1/2]Δz denotes the position of the ith subvolume. Since, in the large-L limit, each subvolume contains a large number, N^iξL2Δz, of microscopic particles, ρ^ does not fluctuate and varies smoothly with position. If the maximal, relative gradient λρ^(z)dρ^dzmgkBT is small compared to λΔz, the density can be considered as approximately constant inside a slab, corresponding to the fine-discretization limit. Thus, the interactions in the coarse-grained model are

Fcg({zβ,qβ})kBT=lnβ=1nVi(rβ)d{η}βVi(rβ)qβ1eH({rβ,{η}β})kBT
(73)
lnβ=1nslabi(zβ)d{ηz}βΔzqβ1eH[ρ^(z)]kBT,withρ^(z)fromEq.(70)
(74)
HFiDis({ρ^i})kBT  withρ^i=ρ^i({zβ,qβ})=λ3L2Δzβ=1nqβδi,i(zβ)
(75)
=Lλ2i=1MΔzλmgzkBTρ^i+v2λ3ρ^i2
(76)
=β=1nqβmgzi(zβ)kBT+v2λ3qβρ^i(zβ).
(77)

Before the fine-discretization approximation, ρ^(z) is calculated from the microscopic configuration, {Rα} = {rβ, {η}β}, according to Eq. (70), whereas the density of microscopic particles in a slab is estimated from the coarse-grained configuration, {zβ, qβ}, according to Eq. (75) afterwards. Within the fine-discretization approximation, the difference between the altitude, zβ, of a coarse-grained particle, β, and the center zi(zβ) of the slab, in which it is located, is negligible, and ρ^(zβ)ρ^i(zβ). Similar expressions hold for the number density of coarse-grained particles, ρ^icg, in the ith slab, ρ^icg1, and ρ^icgq. This completes the description of the grid-based, coarse-grained model within the fine-discretization approximation.

We can exactly calculate the coarse-grained interaction free energy, Fcg({rβ,qβ}), for an ideal gas, v = 0, in the gravitational field and, in  Appendix C, we provide an approximation for finite v,

Fcg({zβ,qβ})kBT=HFiDis({zβ,qβ})kBTβ=1nqβ241+vλ3ρ^i×mgΔzkBT2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)+124mgΔzkBT1+vλ3ρ^i(zβ)2+O(Δz4).
(78)

While this is sufficient to obtain the number of microscopic particles, Ni, in the large subvolumes and to compute the density, ρ^cg(z), of coarse-grained particles, in principle, we have to re-consider the relation between the coarse-grained configuration, {zβ, qβ}, and the microscopic density on scales smaller than a subvolume because the microscopic particles are no longer uniformly distributed inside a subvolume. Therefore, we obtain the local density according to Eq. (52) with a subgrid resolution.

3. Numerical results

(a) Let us start with the example discussed in the Introduction, i.e., an ideal gas, v = 0, of coarse-grained particles that are all comprised of qβ=q¯ microscopic particles, subjected to a gravitational field.

In the limit of fine-discretization, mg(Z0z0)kBTM, the coarse-grained model is described by the free-energy function(al), G({rβ,qβ})=G[ρ^cg(z)],

G[ρ^cg(z)]=Fidcg+Fcg+ΔF,  with  ΔF=FidFidcg
(79)
=F[ρ^(z)=q¯ρ^cg(z)]
(80)
=Lλ2q¯z0Z0dzλρcg(z)lnq¯ρ^cg(z)1+mgzkBTρ^cg(z),
(81)

where the first term arises from the integral over the positions of the coarse-grained particles, the second term models the interaction free energy of the coarse-grained particles, and the third term denotes the additional contributions due to coarse-graining according to Eq. (24). Note that ρ^(r)=q¯ρcg(r) is the density of microscopic particles computed from the coarse-grained configuration, {rβ, qβ}, according to Eq. (52).

In the limit of large subvolumes,

q¯ξΔz=q¯Mξ(Z0z0)L,
(82)

also the coarse-grained model exhibits mean-field behavior, and minimization of G with respect to ρ^cg(z) yields the barometric formula, ρ(z)=q¯ρcg(z)expmgzkBT. This result is confirmed by the simulation of the coarse-grained model with parameters, ξ = 3, mgλkBT=1, v = 0, q¯=2, L=819227λ, and Δz=9λ64, as shown in Fig. 2. For these parameters, q¯ξΔzL2=181.

FIG. 2.

Density distribution of an ideal gas in the gravitational field for q¯=2 and 128 and two spatial discretizations, Δz = λ and 9λ64. Symbols represent the average density, ρ^i, in a subvolume, whereas lines show the density profile with subgrid resolution according to Eq. (52). The data are compared to the barometric formula. For the parameter combination, q¯=128 and Δz=9λ64, pronounced representability problems are visible. The alternate ordinate shows the number of coarse-grained particles n^i=ρ^iVi/q¯. The other parameters are ξ = 3, mgλkBT=1, v = 0, z0 = λ, Z0 = 10λ, and L = 17.418 594λ.

FIG. 2.

Density distribution of an ideal gas in the gravitational field for q¯=2 and 128 and two spatial discretizations, Δz = λ and 9λ64. Symbols represent the average density, ρ^i, in a subvolume, whereas lines show the density profile with subgrid resolution according to Eq. (52). The data are compared to the barometric formula. For the parameter combination, q¯=128 and Δz=9λ64, pronounced representability problems are visible. The alternate ordinate shows the number of coarse-grained particles n^i=ρ^iVi/q¯. The other parameters are ξ = 3, mgλkBT=1, v = 0, z0 = λ, Z0 = 10λ, and L = 17.418 594λ.

Close modal

If we increase the degree of coarse-graining to q¯=128, leaving all other parameters unaltered, the condition, Eq. (82), is not met, q¯ξΔzL2=1. For these parameters, the number, n^i=ρ^icgVi, of coarse-grained particles in a subvolume is small and fluctuates. Thus, (a) the system no longer exhibits mean-field behavior (which is not important for non-interacting particles, v = 0) and (b) the typical configurations of the microscopic model, N^i=0,1,2,, are not well represented by the coarse-grained description, which only allows for N^i=0,q¯,2q¯, according to Eq. (6). The density profile in Fig. 2 exhibits a step-like behavior; each step corresponds to a particular integer value of coarse-grained particles in a subvolume. In this unfavorable case, we also note that the acceptance ratio of random particle displacements of our Monte Carlo simulation significantly decreases compared to q¯=2.

This difference becomes less important when the typical values of n^i are large. For a given overall density, ξ, of microscopic particles and system size, L, this representability problem can only be mitigated by decreasing the number of subvolumes, M, and abandon the fine-discretization approximation. To illustrate this effect, Fig. 2 also presents the density profile for q¯=128 and Δz = λ. Symbols represent the density on the grid, ρ^i, whereas the line shows the density profile, ρ^(z), with subgrid resolution according to Eq. (52). Using Eq. (78), we observe good agreement with the expected behavior even on length scales smaller than Δz.

(b) In order to illustrate the effect of pairwise interactions, we present results for v=12 in Fig. 3. The pairwise repulsion between the particles reduces the compressibility of the gas, and the density decays slower with the altitude, z. The figure presents results for ξ = 3, mgλkBT=1, L=819227λ, and q¯=2 or 128 and Δz=9λ64 or λ. All results agree with the numerical solution of Eq. (72) except the data set with the large degree of coarse-graining, q¯=128, and the fine discretization, Δz=9λ64, which suffers from the representability issue. Increasing Δz and using Eq. (78), we can significantly reduce these artifacts; only for z ≥ 6λ is the density so low that also for the larger Δz representability problems become visible. The good agreement with the other data for z < 6λ, even for the density inside the subvolumes, indicates that the approximation, Eq. (78), of the coarse-grained interaction free energy is rather accurate.

FIG. 3.

Density distribution of a gas of repulsive particles, v = 1/2, in the gravitational field for q¯=2 and 128 and two spatial discretizations, Δz = λ and 9λ64. The symbols and other parameters are identical to Fig. 2.

FIG. 3.

Density distribution of a gas of repulsive particles, v = 1/2, in the gravitational field for q¯=2 and 128 and two spatial discretizations, Δz = λ and 9λ64. The symbols and other parameters are identical to Fig. 2.

Close modal

1. Definition and mean-field behavior

We consider a charged wire with a positive line density, eN/L, of charges along the z direction and negative counterions of charge, −e. The counterions are confined into a cylindrical shell with inner radius, r0, and outer radius, R0. The energy scale is set by kBT, and lengths are measured in units of the Bjerrum length, λ=βe24πϵ. The charge density of counterions is ρe(r)=eρ(r)λ3, where ρ(r) is the dimensionless number density in units of the length scale, λ. The total charge density takes the form

ρtote(r)=eλ3ρ(r)NλLλ2Θr0(r)πr02,
(83)

where Θr0 denotes the index function of the charged cylinder. The line charge density of the wire is quantified by the Manning parameter ξ=NλL, and the overall charge neutrality requires that

0=dr ρtote(r) or N=drλ3ρ(r)or ξ=NλL=2πλ2r0R0drrρ(r)for cylindrical symmetry.
(84)

Similar to the first test system, we consider a system with cylindrical symmetry, i.e., the microscopic particles (counterions) are only characterized by their radial distance, Rα, from the wire. Averaging the density over the azimuthal and z direction, we replace ρ^(r) by

ρ^(r)=λ32πrLα=1Nδ(rRα).
(85)

The assumption that the system is homogeneous in the azimuthal and z direction imparts the mean-field behavior onto the system and asserts that the coarse-grained particles, which have the shape of hollow cylinders, strongly overlap.

We scale the electric potential, Φtot(r), by kBTe and decompose it into the attractive external electric field of the charged wire and the Coulomb repulsion between the like-charged counterions,

ϕtot(r)βeΦtot(r)=ϕext(r)+ϕ^*(r)withϕext(r)=2ξlnrR0,
(86)

where ϕ^* is a linear functional of ρ and therefore depends on the microscopic configuration, {rβ, {η}β}. The charge density and electric potential are related by the Gauss law, λ2ϕ = 4πρ, which in cylindrical geometry adopts the form

rErtotλ=rdϕtotdrr=2ξ+4πλ2r0rdrrρ(r)
(87)

and yields

ϕtot(r)=ϕext(r)drρ(r)λ2|rr|+C
(88)
=2λ2rR0dr2πrρ(r)lnrr,
(89)
Ertot(r)=2λ2rrR0dr2πrρ(r),
(90)

where, in the second-to-last line, we have exploited the cylindrical geometry and adjusted the constant C so that ϕtot(R0)=ϕ^*(R0)=0.

The system is described by the free-energy functional

F[ρ(r)]kBT=drλ3ρ(r)lnρ(r)1ρ(r)ϕext(r)+12ρ(r)drρ(r)λ2|rr|+C
(91)
=drλ3ρ(r)lnρ(r)1ρ(r)ϕext(r)12ρ(r)ϕ^*(r),
(92)
F[ρ(r)]kBT=2πLλdrrλ2ρ(r)lnρ(r)1ρ(r)ϕext(r)12ρ(r)ϕ^*(r).
(93)

The prefactor, λ/L, plays the role of the Ginzburg parameter so that the system exhibits the mean-field behavior in the limit L. Thus, the equilibrium density profile is obtained by minimizing F[ρ(r)] with the constraint that the number of counterions is N, which yields together with Eq. (89) the Poisson-Boltzmann equation

ln(1+γ2)λ22πR02=lnρρϕext+ϕ=lnρρϕtot.
(94)

Here we have used that the electric potential ϕ^* is a linear functional of the density; see Eq. (88). The chemical potential on the left-hand side contains the Lagrange multiplier, γ, of the constraint number of counterions. The analytic solution of the Poisson-Boltzmann equation, Eqs. (89) and (94), is given by68 

ϕtot(r)=2ln1+γ2γrR0cosγlnrRMwithRMr0=exp1γatan1ξγ,
(95)
ρ(r)=(1+γ2)λ22πR02exp[ϕtot(r)],
(96)
Ertot(r)=2λr1γtanγlnrRM,
(97)

where the Lagrange multiplier, γ, is numerically determined from

γlnr0R0atan1ξγ+atan1γ=0.
(98)

These equations provide an explicit, analytical solution of the mean-field behavior that serves as a reference to validate our simulations. In particular, we note that if we scale all lengths, r0, r, and R0, by a factor of Λ at fixed ξ, the density of counterions scales like ρ(r) → ρ′(r) = ρr)/Λ2.

2. Coarse-grained simulations—Using hollow cylinders for grid-based and off-lattice models

We discretize the radial dimension, r0rR0, in M hollow cylinders of constant thickness, Δr=R0r0M, and length L. The central radius of the ith subvolume is ri = r0 + [i − 1/2]Δr, and its volume is given by Vi = 2πriΔrL.

Using hollow long cylinders as subvolumes, (a) we enforce the mean-field behavior because the Ginzburg parameter is small, λ/L → 0 for long cylinders, and (b) we mitigate representability issues because the number, Ni, of counterions in each subvolume, Ni = 2πriΔrLρi, remains large for L even if we choose Δr small enough for the approximation of fine discretization to be accurate.

In the large-L limit, each subvolume contains a large number of counterions, N^iL so that ρ^(r) is a smooth function of r for each coarse-grained configuration. The inverse characteristic length scale of the system is set by69 

λρ(r)dρdr=λdϕtotdr2ξλr0.
(99)

Thus, the limit of fine discretization is reached for M2ξR0r01, and we obtain

Fcg({zβ,qβ})kBTFiDisH({ρ^i})kBTwith   ρ^i=ρ^i({rβ,qβ})=λ32πriΔrLβ=1nqβδi,i(rβ)
(100)
=2πLλi=1MΔrriλ2ρ^iϕext(ri)+12ρ^iϕ^*(ri)
(101)
=β=1nqβϕext(ri(rβ))+12qβϕ^*(ri(rβ)).
(102)

The densities of coarse-grained particles and those with charges qβ = 1 and qβ = q are computed in analogy to Eq. (100). The partition function of the coarse-grained model can be re-written in terms of the partition function of particles with a one-dimensional coordinate, rβ, and charge, qβ,

Z=1n!{qβ}β=1nr0R0drβ2πrLλ3eFcg({rβ,qβ})+F(0)TSQ+FidFidcgkBT
(103)
=1n!{qβ}β=1nr0R0drβλeFcg({rβ,qβ})+Fjac({rβ})+F(0)TSQ+FidFidcgkBT
(104)

with

Fjac({rβ})kBT=β=1nln2πrβLλ2,
(105)

where the additional free energy, Fjac({rβ}), accounts for the Jacobian of the transformation from cylindrical coordinates to a one-dimensional radial coordinate. This completes the description of the grid-based coarse-grained model with hollow cylindrical subvolumes within the approximation of fine discretization.

An approximate off-lattice model is obtained by defining the one-dimensional density, ρ^(rβ), of charges around the coarse-grained particle, β, via

ρ^rβ=β=1nqβw(rβ,rβ)
(106)
withw(rβ,rβ)=λ32πr̃βΔrLif |rβr̃β|<Δr20,otherwiseandr̃β=r0+Δr2if rβ<r0+Δr2rβif r0+Δr2rβR0Δr2R0Δr2if rβ>R0Δr2.

The shift of the central radius, rβr̃β, of the hollow cylinder, which characterizes the shape of the particle, in the vicinity of the hard impenetrable walls mitigates the missing-neighbor effect. Note that in the cylindrical symmetry, the volume of a particle, ΔV(rβ)=2πr̃βΔrL, in Eq. (60) depends on the position of the coarse-grained particle and is also affected by the shift of the central radius.

3. Numerical results

For the numerical study, we use the Manning parameter ξ = 3. The system geometry is characterized by the inner radius, r0 = λ, the outer radius, R0 = 10λ, and the cylinder length, L=81923λ. The total number of microscopic particles (counterions) is N=ξLλ=8192. We use M = 64 hollow cylinders as subvolumes, i.e., Δr=9λ64.

In Fig. 4, we compare the counterion density, ⟨ρ(r)⟩, with the prediction of Poisson-Boltzmann theory for a small and large degree of coarse-graining. For the former case, we keep the ratio N/n = 2 constant, i.e., the average degree of coarse-graining is 2, and we consider various choices of a priori distributions, PnM{Ni}(0)({ni}) and pNini(0)({qiβi}), and also consider the off-lattice implementation. Red circles correspond to constant q¯=2 (according to Sec. II D 1). Filled circles refer to an array of cylindrical subvolumes, whereas open circles present the results of the off-lattice approximation. Blue squares present the result of a fluctuating q according to Eq. (49) using cylindrical subvolumes (filled symbols) or the off-lattice approximation (open squares), respectively. Black diamonds show the counterion density for a bimodal distribution, q = 1 and 3, of counterions per coarse-grained particle, using Eq. (47). In order not to bias the spatial distribution of coarse-grained particles, we have used Eq. (31), and the corresponding results are presented by filled diamonds. Alternatively, we have also employed Eq. (34) with a spatially varying ϵ in order to increase the number of coarse-grained particles with no degree of coarse-graining, q = 1, in the vicinity of the charged wire. The open diamonds show the corresponding counterion density. Triangles present the results for a large constant degree of coarse-graining, q¯=128. Additionally, the figure depicts the result for fluid elements, see Eq. (42), by crosses.

FIG. 4.

Comparison of the counterion density of the coarse-grained model (symbols) for various coarse-graining schemes with the exact prediction of the Poisson-Boltzmann theory (solid line). Cylindrical symmetry is imposed. ξ = 3, r0 = λ, R0 = 10λ, and all systems are comprised of 8192 microscopic particles (counterions).

FIG. 4.

Comparison of the counterion density of the coarse-grained model (symbols) for various coarse-graining schemes with the exact prediction of the Poisson-Boltzmann theory (solid line). Cylindrical symmetry is imposed. ξ = 3, r0 = λ, R0 = 10λ, and all systems are comprised of 8192 microscopic particles (counterions).

Close modal

For all combinations, we find good quantitative agreement, which is expected because the Poisson-Boltzmann approach becomes correct in the quasi-one-dimensional cylindrical system. Only for q¯=128, we find a small “step” in the density profile around r ≈ 2λ, which indicates the representability problem at these low densities similar to what we discussed in Sec. III A 3.

Whereas the different coarse-graining schemes give rise to the same physical distribution of microscopic particles, the profiles, ⟨ρcg(r)⟩, of the coarse-grained particles, shown in Fig. 5, markedly differ.

FIG. 5.

Radial distribution of the density distribution of coarse-grained particles. Symbols correspond to different coarse-graining schemes (see Fig. 4). Since for most schemes the average degree of coarse-graining is 2, we compare the data to half the density of counterions according to the Poisson-Boltzmann theory (solid line).

FIG. 5.

Radial distribution of the density distribution of coarse-grained particles. Symbols correspond to different coarse-graining schemes (see Fig. 4). Since for most schemes the average degree of coarse-graining is 2, we compare the data to half the density of counterions according to the Poisson-Boltzmann theory (solid line).

Close modal

For the case, qβ=q¯, of course, the distribution of coarse-grained particles follows from ρ^icg=ρ^i/q¯. Also for the a priori distribution, PnM{Ni}(0)({ni}), according to Eq. (34) with ϵi = 1 for all i = 1, , M, the ratio ⟨ρ(r)⟩/⟨ρcg(r)⟩ ≈ 2 is independent from the spatial position. This observation also holds true for a bimodal distribution of counterions per coarse-grained particle, qβ = 1 or 3, which corresponds to the a priori distribution, pNini(0)({qiβi}), according to Eq. (47), or for a fluctuating number of counterions per coarse-grained particle, qβ = 0, 1, , which corresponds to the a priori distribution, pNini(0)({qiβi}), according to Eq. (49).

In the bimodal case with ϵi = 1, the distribution of each coarse-grained species, q = 1 and q = 3, is proportional to ⟨ρ(r)⟩. For our specific choice, qβ = 1 and 3, the number of both types of coarse-grained particles is the same. Thus, their distributions agree, i.e., 2⟨ρcg1(r)⟩ ≈ 2⟨ρcgq(r)⟩ = ⟨ρ(r)⟩/2.

We control the distribution of coarse-grained particles by varying ϵi in the different subvolumes [according to Eq. (34)]. In these Monte Carlo simulation, we use random particle displacements and a Monte Carlo move that swaps the contents of two, randomly selected particles, i and j, for qiqj. Both Monte Carlo moves are subjected to a Metropolis acceptance criterion according to the equilibrium distribution of the coarse-grained model. The non-locality of the latter Monte Carlo move prevents a dynamical interpretation of the sequence of states, but it allows for a very efficient equilibration of the microscopic (charge) density. More sophisticated algorithms that locally make, break, and remake clusters of particles, like the “SWINGER” algorithm of Praprotnik and co-workers,42 could be advantageous if an appropriate acceptance criterion was available.

As indicated by the stars in Fig. 5 (referring to the right ordinate axis), large values of ϵi = 3 are located in the vicinity of the charged wire and favor the presence of (not) coarse-grained particles with qβ = 1, i.e., microscopic particles. Further away from the central wire, ϵi adopt the value 1 and promote a large degree of coarse-graining. Figure 5 demonstrates that in the region, ϵi = 3, the density of q = 1-particles is more than 3 times larger than that of q = 3-particles, and the suppression of the density of q = 3-particles at the wire could be increased even more by increasing the value of ϵi. In the volume with ϵi = 1, in turn, the density of q = 1-particles is vanishingly small and the system is entirely represented by coarse-grained q = 3-particles. Such a switch of degree of coarse-graining as a function of the distance from an embedded object (wire, surface, macromolecule) may be useful in circumstances where it is difficult to calculate the effective excess free energy, Fcg, that describes the interaction of the coarse-grained particles among each other and with the embedded object (e.g., because of strong liquid-like layering effects), yet an accurate coarse-grained description is available farther away from the embedded object.70 In the present example, Fcg is only given by the electrostatic energy, but more sophisticated schemes to devise the effective interactions between particles with different degree of coarse-graining are available.71,72

We have presented a general formalism of coarse-graining where multiple, indistinguishable, microscopic particles are stochastically mapped into one effective, coarse-grained interaction center. In addition to devising appropriate coarse-grained interactions, Fcg, see Eq. (16),21–23,25–27 one also has to account for the loss of translational entropy associated with the decimation of indistinguishable particles.

We have considered different schemes of coarse-graining, see Table I, in which a coarse-grained particle, β, is comprised of a fixed number, q¯, of microscopic particles (i.e., 1-to-q¯ mapping) or is characterized by an internal variable, qβ, that quantifies its fluctuating content of microscopic particles. We have discussed schemes where qβ fluctuates around a local degree of coarse-graining, q¯i, or where the density of coarse-grained particles (“fluid elements”) is independent from that of the microscopic ones.

In all schemes, the coarse-graining of translational entropy gives rise to density-dependent, effective interactions on the coarse-grained scale. Dividing the system volume into non-overlapping subvolumes to facilitate counting, we have derived explicit expressions for the coarse-grained interactions, Fcg, and the additional coarse-grained entropic contribution to the interaction free energy on the coarse-grained level. Moreover, we have indicated how this exact, grid-based procedure can be generalized to an approximate off-lattice description using density-dependent but translationally invariant interactions. These density-dependent interactions are particularly important in systems with pronounced spatial inhomogeneities.

Using two simple examples, an ideal gas in a gravitational field and distribution of counterions around a linear charged wire within mean-field approximation, we have illustrated our scheme and validated it against exact, analytical results. We expect that this density-dependent, coarse-grained entropy be useful for devising coarse-grained models for solvents in biophysical simulations or ionic liquids.

It is a great pleasure to thank William Noid and Matej Praprotnik for stimulating and encouraging comments on this manuscript. Financial support has been provided by the German Science Foundation (DFG) SFB 1073/TP A03. J.J.dP. and N.E.J. gratefully acknowledge support from the U.S. Department of Energy Office of Science, Program in Basic Energy Sciences, Materials Sciences and Engineering Division. The calculations have been performed at the GWDG Göttingen, HLRN Hannover/Berlin, Neumann Institute for Computing, Jülich, Germany. N.E.J. thanks the Argonne National Laboratory Maria Goeppert Mayer Named Fellowship for support.

Starting from Eq. (14) with H=0, we derive the partition function of an ideal gas,

Zid=1n!{qβ}β=1nVdrβλ3PnM{N^i}(0)({n^i})×ei=1MlnN^i!λ3ViN^ilnn^i!λ3Vin^ilnpN^in^i(0)({qiβi})=1λ3N{qβ}1n!β=1nVdrβPnM{N^i}(0)({n^i})×i=1Mn^i!ViN^iN^i!Vin^ipN^in^i(0)({qiβi}).
(A1)

Note that the n^i depend on the positions of the coarse-grained particles (as indicated by “ ˆ ”). Instead of integrating the coarse-grained particle positions over the entire volume V, we first distribute the coarse-grained particles into the subvolumes and subsequently integrate over their positions inside this subvolume, i.e.,

1n!β=1nVdrβ={ni}i=1M1ni!βi=1niVidriβi.
(A2)

Since the particle i is localized in subvolume Vi and comprises qiβi microscopic particles, the number of coarse-grained and microscopic particles in the subvolume Vi are given by n^iβ=1Nδi,i(rβ)=ni and N^iβ=1nqβδi,i(rβ)=βi=1niqiβi=Ni, respectively. Note that {qβ}={Ni}i=1M{qiβi} because the sum {qiβi}=qi1qiniδNi,βi=1niqiβi restricts the number of microscopic particles in each subvolume Vi to Ni.

Thus, we can perform the integration over the explicit particle coordinates, yielding

Zid=1λ3N{ni}i=1M1ni!Vini×{Ni}PnM{N^i}(0)({n^i})i=1M{qiβi}ni!ViNiNi!VinipNini(0)({qiβi})=1λ3N{ni}{Ni}PnM{N^i}(0)({n^i})×i=1MViNiNi!{qiβi}pNini(0)({qiβi})=1{ni}and{Ni}
(A3)
=1λ3NN!{ni}{Ni}PnM{N^i}(0)({n^i})×N!i=1MNi!i=1MViNi
(A4)
=1λ3NN!{Ni}{ni}PnM{N^i}(0)({n^i})=1{Ni}×N!i=1MNi!i=1MViNi
(A5)
=1λ3NN!i=1MViN=VNλ3NN!.
(A6)

Alternatively and equivalently, we can conceive this Monte Carlo move of a coarse-grained particle as a cluster move in the original system of microscopic particles. To this end, we randomly select a microscopic particle from the N=q¯n microscopic particles in the system. The probability that this selected particle is identical to the one that defines the explicit DoF of the coarse-grained particle β* is 1/(q¯n). This particle is located in subvolume Viold, which contains a total of Niold=q¯niold microscopic particles. The number of possibilities to add q¯1 other microscopic particles from this subvolume, Viold, to the cluster is

q¯niold1q¯1=(q¯niold1)!(q¯1)!(q¯[niold1])!=q¯q¯niold(q¯niold)!q¯!(q¯[niold1])!=1nioldq¯nioldq¯  .
(B1)

The probability to construct the specific set of microscopic particles that make up the coarse-grained particle, β*, is the inverse of that number. Subsequently we choose a displacement vector of the first microscopic particle of the cluster according to pΔr). Like in the Monte Carlo move of the coarse-grained model, the probability to propose a specific new set of the q¯1 microscopic particle coordinates, {η}β*, in the new subvolume is 1/Vinewq¯1. Thus, the overall probability to propose this cluster move in the original system of microscopic particles is

poldnewprop=1q¯n11nioldq¯nioldq¯pΔ(Δr)1Vinewq¯1.
(B2)

Within the original system of microscopic particles, the ratio of the statistical weights of the new and old configuration simply is

pneweqpoldeq=eΔHkBT.
(B3)

Thus, the probability to accept the Monte Carlo move in the original system of microscopic particles is

poldnewacc=min1,pneweqpoldeqpnewoldproppoldnewprop=min1,1q¯npΔ(Δr)Violdq¯11ninewq¯ninewq¯1q¯npΔ(Δr)Vinewq¯11nioldq¯nioldq¯eΔHkBT,
(B4)

which, gratifyingly, agrees with Eq. (27).

In order to calculate the coarse-grained interaction free energy, Fcg, for the interacting particles in a gravitational field, we assume that the microscopic density profile inside the ith subvolume (slab) centered at zi is approximately linear, i.e., the gravitational field gives rise to a inhomogeneous density distribution (“polarization”) inside of a subvolume.

We employ the fluctuations, δρ^(z),

δρ^(z)i=1Mρ^(z)ρ^i+γiλ(zzi)ΘVi(z)
(C1)

around this linear profile as small parameter. ΘVi(r) denotes the index function of the subvolume, Vi. Vidrδρ^(z)=0 and Vidr(zzi)=0 in all subvolumes. Here, γi/λ with i = 1, , M are variational parameters that quantify the local slope of the density profile. ρ^(z) denotes the density of microscopic particles according to Eq. (51), whereas ρ^i is the average density of microscopic particles in the ith subvolume, Eq. (54).

The interactions comprise the external gravitational field and the repulsive binary interactions. The Hamiltonian of the binary interactions takes the form

Hv[ρ^]kBT=v2λ3Vdrλ3ρ^2(z)
(C2)
=v2λ3i=1MVidrλ3ρ^i2+γi2λ2(zzi)2+δρ^2(z)2ρ^iγiλ(zzi)+2ρ^iδρ^(z)2γiλ(zzi)δρ^(z)=v2λ3i=1MVidrλ3ρ^i2+γi2λ2(zzi)2+δρ^2(z)2γiλ(zzi)ρ^(z)ρ^i+γiλ(zzi)
(C3)
=v2λ3i=1MVidrλ3ρ^i2γi2λ2(zzi)2+δρ^2(z)2γiλ2(zzi)ρ^(z)
(C4)
=v2λ3i=1MViλ3ρ^i2γi2λ2L2λ323Δz23+Vidrλ32γiλ(zzi)ρ^(z)+δρ^2(z)
(C5)
=β=1nv2λ3qβρ^i(rβ)v24λ3Δz2λ2i=1MViλ3γi2+i=1MVidrλ3vγiλ4(zzi)ρ^(z)+vδρ^2(z)2λ3,
(C6)

and we obtain for the total Hamiltonian

H[ρ^]=Vdrλ3mgzρ^(z)+vkBT2λ3ρ^2(z)
(C7)
=i=1MViλ3mgziρ^i+i=1MVidrλ3mg(zzi)ρ^(z)+vkBT2λ3ρ^2(z)
(C8)
=β=1nmgzi(rβ)+vkBT2λ3qβρ^i(rβ)=HFiDis({rβ,qβ})vkBT24λ3Δz2λ2i=1MViλ3γi2+i=1MVidrλ3mgvkBTγiλ4(zzi)ρ^(z)+Vdrλ3vkBTδρ^2(z)2λ3
(C9)
=HFiDis({rβ,qβ})vkBT24λ3Δz2λ2i=1MViλ3γi2+β=1|qβ>0nmgvkBTγi(rβ)λ4(zβzi(rβ))+αβ=1qβ1(ηzαβzi(rβ))+Vdrλ3vkBTδρ^2(z)2λ3.
(C10)

Neglecting the last term, which is proportional to the fluctuations of the microscopic density profile around the linear, mean-field approximation, we are able to compute the coarse-grained interaction free energy, Fcg({zβ,qβ}), because the internal, positional DoF decouple. Introducing an effective, dimensionless, external field, himgλkTvλ3γi, in the ith subvolume, we obtain

eFcgHFiDis+vkBT24λ3Δz2λ2i=1MViλ3γi2kBTβ=1|qβ>0nVi(rβ)d{η}βVi(rβ)q¯1ehi(rβ)zβzi(rβ)+αβ=1q¯1ηzαβzi(rβ)λ
(C11)
=β=1|qβ>0nehi(rβ)zβzi(rβ)λαβ=1qβ1zi(zβ)Δz/2zi(zβ)+Δz/2dηzαβΔzehi(rβ)ηzαβzi(rβ)λ=β=1|qβ>0nehi(rβ)zβzi(rβ)λsinhhi(rβ)Δz2λhi(rβ)Δz2λqβ1,
(C12)
Fcg({zβ,qβ})kBT=HFiDis({zβ,qβ})kBTv24λ3Δz2λ2i=1MViλ3γi2+β=1|qβ>0nhi(rβ)zβzi(zβ)λ(qβ1)lnsinhhi(rβ)Δz2λhi(rβ)Δz2λ.
(C13)

Note that this coarse-grained interaction free energy, Fcg({rβ,qβ}), is the only term in the statistical weight of the coarse-grained configuration that explicitly depends on the position of the coarse-grained particle, rβ, inside of a subvolume; all other contributions are functions of the number of microscopic or coarse-grained particles, ρ^i(rβ) and ρ^i(rβ)cg, respectively. Thus, we define an averaged coarse-grained free-energy, F¯cg, by integrating rβ over Vi(rβ),

F¯cgkBTlnβ=1nVi(rβ)drβVi(rβ)×eFcg({zβ,qβ})kBT
(C14)
=HFiDis({zβ,qβ})kBTv24λ3Δz2λ2i=1MViλ3γi2β=1nqβlnsinhhi(rβ)Δz2λhi(rβ)Δz2λ
(C15)
=HFiDis({zβ,qβ})kBTv24λ3Δz2λ2i=1MViλ3γi2β=1nqβ16hi(rβ)Δz2λ2+O(Δz4)
(C16)
=HFiDis({zβ,qβ})kBTΔz224λ2i=1MViλ3vλ3γi2Δz224λ2β=1nqβhi(rβ)2+O(Δz4),
(C17)

which explicitly demonstrates that the terms beyond the fine-discretization approximation are of order Δz2. In order to determine the variational parameters, γi, we use the expression up to order Δz2 and compute the optimal values, γi*,

1kBTF¯cgγiΔz224λ2Viλ3vλ32γiΔz224λ2ρ^iVi2hi(rβ)vλ3
(C18)
=Δz212λ2Viλ3vλ3γi+ρ^imgλkTvλ3γi
(C19)
=Δz212λ2Viλ3vλ3ρ^iγi1ρi+vλ3+mgλkT=!0,
(C20)
γi*=mgλkT1ρi+vλ3=mgλρ^ikBT1+vλ3ρ^i,
(C21)
hi*=mgλkTvγi*λ3=mgλkBT1+vλ3ρ^i.
(C22)

Thus, the gradient, −γi/λ, results from the linear response of the gas with an isothermal compressibility that is proportional to 1ρi+vλ3, to the gravitational field, mgλkT.

Alternatively, we obtain the gradient of the density profile by differentiating Eq. (72) with respect to z,

0=1kBTdμdz=1ρdρdz+mgkBT+vλ3dρdz=1ρ+vλ3γλ+mgkBT.
(C23)

Since both approaches neglect fluctuations, they yield the identical result.

Inserting the optimal values, γi* and hi*, we obtain the coarse-grained interaction free-energy

Fcg({zβ,qβ})kBT=HFiDis({zβ,qβ})kBTv24λ3Δz2λ2i=1MViλ3mgλρ^ikBT1+vλ3ρ^i2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)(qβ1)lnsinhmgΔz2kBT1+vλ3ρ^i(zβ)mgΔz2kBT1+vλ3ρ^i(zβ)
(C24)
=HFiDis({zβ,qβ})kBTβ=1nqβ16vλ3ρ^i(zβ)mgΔz2kBT1+vλ3ρ^i2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)(qβ1)lnsinhmgΔz2kBT1+vλ3ρ^i(zβ)mgΔz2kBT1+vλ3ρ^i(zβ)
(C25)
=HFiDis({zβ,qβ})kBTβ=1nqβ16vλ3ρ^i(zβ)mgΔz2kBT1+vλ3ρ^i2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)(qβ1)16mgΔz2kBT1+vλ3ρ^i(zβ)2+O(Δz4)=HFiDis({zβ,qβ})kBTβ=1nqβ16vλ3ρ^i(zβ)mgΔz2kBT1+vλ3ρ^i2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)(qβ1)16mgΔz2kBT1+vλ3ρ^i(zβ)2+O(Δz4)=HFiDis({zβ,qβ})kBTβ=1nqβ241+vλ3ρ^imgΔzkBT2+β=1|qβ>0nmgzβzi(zβ)kBT1+vλ3ρ^i(zβ)+124mgΔzkBT1+vλ3ρ^i(zβ)2+O(Δz4).
(C26)

For an ideal gas, v = 0 in the gravitational field, the effective coarse-grained interaction, Fcg, in Eq. (C13) is exact and does not depend on γi, i.e.,

Fcg({zβ,qβ})=β=1|qβ>0nmgzβ+(q¯1)zi(zβ)+Cwith C=(q¯1)nkBTlnsinhmgΔz2kBTmgΔz2kBT,
(D1)

where the constant, C, does not depend on the configuration, {zβ, qβ}, of the coarse-grained model. This simpler expression allows us to explicitly relate the partition function of the coarse-grained model for large subvolumes and the partition function of the original microscopic model

Zcg=1n!β=1nVdrβλ3eβ=1nmgzβ+(q¯1)zi(zβ)kBTsinhmgΔz2kBTmgΔz2kBT(q¯1)nei=1Mln(q¯n^i)!λ3Viq¯n^ilnn^i!λ3Vin^i
(D2)
=1n!{ni}n!i=1Mni!i=1Mβi=1niVidriβiλ3ei=1Mmgβi=1niziβi+(q¯1)nizikBTsinhmgΔz2kBTmgΔz2kBT(q¯1)nei=1Mln(q¯ni)!λ3Viq¯nilnni!λ3Vini
(D3)
={ni}i=1MViλ3nii=1Mni!i=1Mβi=1niVidriβiViemgziβikBTei=1Mmg(q¯1)nizikBTsinhmgΔz2kBTmgΔz2kBT(q¯1)nei=1Mln(q¯ni)!λ3Viq¯nilnni!λ3Vini
(D4)
={ni}ei=1Mmgq¯nizikBTsinhmgΔz2kBTmgΔz2kBTq¯nei=1Mln(q¯ni)!λ3Viq¯ni
(D5)
=sinhmgΔz2kBTmgΔz2kBTq¯n{ni}i=1M1(q¯ni)!Viλ3q¯niemgq¯nizikBT
(D6)
={Ni=q¯ni}i=1M1Ni!Viλ3NiemgzikBTsinhmgΔz2kBTmgΔz2kBTNi
(D7)
=1N!{Ni=q¯ni}N!i=1MNi!i=1MViαi=1NidRiαiλ3emgZiαikBT.
(D8)

This partition function would be the partition function of the original microscopic model if the sum ran over all values, Ni = 0, 1, 2, . Instead, the result is the partition function obtained by summing over all microscopic configurations that can be represented in the form {rβ, {η}β}, i.e., Ni=0,q¯,2q¯,.

If n^i1 in the typical configurations of the coarse-grained model, we can approximate ni=0,1,dni=1q¯dNi1q¯Ni=0,1,. Thus, if we choose the discretization coarse enough for each subvolume to contain many coarse-grained particles, i.e., the coarse-grained particles strongly overlap, we can approximately relate the partition function of the coarse-grained model, Zcg, and the original, microscopic model, Z,

Zcg1N!1q¯M{Ni}N!i=1MNi!i=1MViαi=1NidRiαiλ3emgZiαikBT
(D9)
=1q¯M1N!Vα=1NdRαλ3emgZiαikBT=1q¯M×Z,
(D10)

where the factor q¯M accounts for the ratio of the number of configurations of the original model of microscopic particles and its coarse-grained representation.

1.
P. G.
de Gennes
,
Phys. Lett. A
38
,
339
(
1972
).
2.
J. F.
Douglas
and
K. F.
Freed
,
Macromolecules
17
,
2344
(
1984
).
3.
J.
Baschnagel
,
K.
Binder
,
P.
Doruker
,
A. A.
Gusev
,
O.
Hahn
,
K.
Kremer
,
W. L.
Mattice
,
F.
Müller-Plathe
,
M.
Murat
,
W.
Paul
 et al,
Adv. Polym. Sci.
152
,
41
(
2000
).
5.
M.
Praprotnik
,
L.
delle Site
, and
K.
Kremer
,
Annu. Rev. Phys. Chem.
59
,
545
(
2008
).
6.
G.
Ayton
and
G.
Voth
,
Biophys. J.
83
,
3357
(
2002
).
7.
M.
Müller
,
K.
Katsov
, and
M.
Schick
,
J. Polym. Sci., Part B: Polym. Phys.
41
,
1441
(
2003
).
8.
S. J.
Marrink
,
H. J.
Risselada
,
S.
Yefimov
,
D. P.
Tieleman
, and
A. H.
de Vries
,
J. Phys. Chem. B
111
,
7812
(
2007
).
9.
T.
Murtola
,
A.
Bunker
,
I.
Vattulainen
,
M.
Deserno
, and
M.
Karttunen
,
Phys. Chem. Chem. Phys.
11
,
1869
(
2009
).
10.
M.
Hömberg
and
M.
Müller
,
J. Chem. Phys.
132
,
155104
(
2010
).
11.
Z. J.
Wang
and
M.
Deserno
,
New J. Phys.
12
,
095004
(
2010
).
12.
T. A.
Knotts
,
N.
Rathore
,
D. C.
Schwartz
, and
J. J.
de Pablo
,
J. Chem. Phys.
126
,
084901
(
2007
).
13.
N. B.
Becker
and
R.
Everaers
,
Phys. Rev. E
76
,
021923
(
2007
).
14.
Z. Y.
Zhang
,
L. Y.
Lu
,
W. G.
Noid
,
V.
Krishna
,
J.
Pfaendtner
, and
G. A.
Voth
,
Biophys. J.
95
,
5073
(
2008
).
15.
W. G.
Noid
,
J. Chem. Phys.
139
,
090901
(
2013
).
16.
A. A.
Louis
,
J. Phys.: Condens. Matter
14
,
9187
(
2002
).
17.
E. C.
Allen
and
G. C.
Rutledge
,
J. Chem. Phys.
128
,
154115
(
2008
).
18.
S.
Izvekov
,
P. W.
Chung
, and
B. M.
Rice
,
J. Chem. Phys.
135
,
044112
(
2011
).
19.
M.
Fukuda
,
H.
Zhang
,
T.
Ishiguro
,
K.
Fukuzawa
, and
S.
Itoh
,
J. Chem. Phys.
139
,
054901
(
2013
).
20.
N. J. H.
Dunn
,
T. T.
Foley
, and
W. G.
Noid
,
Acc. Chem. Res.
49
,
2832
(
2016
).
21.
D.
Reith
,
M.
Pütz
, and
F.
Müller-Plathe
,
J. Comput. Chem.
24
,
1624
(
2003
).
23.
A. P.
Lyubartsev
and
A.
Laaksonen
,
Phys. Rev. E
52
,
3730
(
1995
).
24.
A. P.
Lyubartsev
,
M.
Karttunen
,
I.
Vattulainen
, and
A.
Laaksonen
,
Soft Mater.
1
,
121
(
2002
).
25.
S.
Izvekov
,
M.
Parrinello
,
C. J.
Burnham
, and
G. A.
Voth
,
J. Chem. Phys.
120
,
10896
(
2004
).
26.
M. S.
Shell
,
J. Chem. Phys.
129
,
144108
(
2008
).
27.
T. T.
Foley
,
M. S.
Shell
, and
W. G.
Noid
,
J. Chem. Phys.
143
,
243104
(
2015
).
28.
M.
Schöberl
,
N.
Zabaras
, and
P.-S.
Koutsourelakis
,
J. Comput. Phys.
333
,
49
(
2017
).
29.
P.
Espanol
and
P. B.
Warren
,
J. Chem. Phys.
146
,
150901
(
2017
).
30.
E. G.
Flekkøy
and
P. V.
Coveney
,
Phys. Rev. Lett.
83
,
1775
1778
(
1999
).
31.
M.
Serrano
,
G.
de Fabritiis
,
P.
Espanol
,
E. G.
Flekkoy
, and
P. V.
Coveney
,
J. Phys. A: Math. Gen.
35
,
1605
1625
(
2002
).
32.
P.
Espanol
and
M.
Revenga
,
Phys. Rev. E
67
,
026705
(
2003
).
33.
J. B.
Avalos
and
A. D.
Mackie
,
Europhys. Lett.
40
,
141
(
1997
).
34.
P.
Espanol
,
Europhys. Lett.
40
,
631
(
1997
).
35.
M.
Langenberg
and
M.
Müller
,
Europhys. Lett.
114
,
20001
(
2016
).
36.
G.
Stoltz
,
J. Comput. Phys.
340
,
451
(
2017
).
37.
J. F.
Dama
,
A. V.
Sinitskiy
,
M.
McCullagh
,
J.
Weare
,
B.
Roux
,
A. R.
Dinner
, and
G. A.
Voth
,
J. Chem. Theory Comput.
9
,
2466
(
2013
).
38.
H.
Bock
,
K. E.
Gubbins
, and
S. H. L.
Klapp
,
Phys. Rev. Lett.
98
,
267801
(
2007
).
39.
J. M.
Drouffe
,
A. C.
Maggs
, and
S.
Leibler
,
Science
254
,
1353
(
1991
).
40.
T.
Sanyal
and
M. S.
Shell
,
J. Chem. Phys.
145
,
034109
(
2016
).
41.
S. M.
Gopal
,
A. B.
Kuhn
, and
L. V.
Schäfer
,
Phys. Chem. Chem. Phys.
17
,
8393
(
2015
).
42.
J.
Zavadlav
,
S. J.
Marrink
, and
M.
Praprotnik
,
J. Chem. Theory Comput.
12
,
4138
(
2016
).
43.
M.
Deng
,
Z.
Li
,
O.
Borodin
, and
G. E.
Karniadakis
,
J. Chem. Phys.
145
,
144109
(
2016
).
44.
M. R.
Delyser
and
W. G.
Noid
,
J. Chem. Phys.
147
,
134111
(
2017
).
45.
G.
Faure
,
J.-B.
Maillet
, and
G.
Stoltz
,
J. Chem. Phys.
140
,
114105
(
2014
).
46.
M.
Müller
and
N. B.
Wilding
,
Phys. Rev. E
51
,
2079
(
1995
).
47.
J. P.
Wittmer
,
A.
Milchev
, and
M. E.
Cates
,
J. Chem. Phys.
109
,
834
(
1998
).
48.
K. C.
Daoulas
,
A.
Cavallo
,
R.
Shenhar
, and
M.
Müller
,
Soft Matter
5
,
4499
(
2009
).
49.
N. B.
Wilding
and
P.
Sollich
,
J. Chem. Phys.
116
,
7116
(
2002
).
50.
M.
Buzzacchi
,
P.
Sollich
,
N. B.
Wilding
, and
M.
Müller
,
Phys. Rev. E
73
,
046110
(
2006
).
51.
M.
Müller
,
J. Stat. Phys.
145
,
967
(
2011
).
52.

In the approximate off-lattice description in Sec. II F, the “size” is the range of the weighting function, w, that defines the weighted density, Eq. (60).

53.

An internal, positional DoF refers to d scalar degrees of freedom where d denotes the spatial dimension.

54.

Obviously, this constraint on the a priori distribution becomes irrelevant if there is only a single subvolume, M = 1 and V = V1.

55.

This constraint is somewhat similar to coarse-graining along the backbone of a molecule where the decimated, internal DoFs are also restrained around the positions of the coarse-grained particles, which define the conformation of the molecule.

56.

In addition to one-body terms, we expect Fcg also to include two-body terms (and higher-order interactions), in contrast to coarse-graining strategies where the DoFs associated with a coarse-grained particle are delocalized in the entire system.38 The position of the coarse-grained particle, rβ, coincides with the position of a microscopic particle, cf. Eq. (5).

57.
J. W.
Wagner
,
T.
Dannenhoffer-Lafage
,
J.
Jin
, and
G. A.
Voth
,
J. Chem. Phys.
147
,
044113
(
2017
).
58.

The special, trivial case, q¯=1, corresponds to the original system of coarse-grained particles.

59.

Note that this expression for F(0) does not assume that there are many microscopic or coarse-grained particles in a subvolume.

60.

Since qβ = 0 is allowed, there can be subvolumes that contain a coarse-grained particle but no microscopic particle, i.e., Ni = 0 but ni > 0. In these subvolumes, we set Zi(Ni)=λ3Vi, which excludes these subvolumes from the last sum in Eq. (30).

61.

Given N and n, however, the number of coarse-grained particles with qβ = 1 and qβ = q in the entire system is fixed.

62.

In the general case, we approximateH({Rα})=H({rβ,{η}β})=H[ρ^(r)]Fcg({rβ,qβ}),where the free-energy functional, Fcg, accounts for the local, packing correlations because the length scale of the interaction between the microscopic particles is not much smaller than the linear dimension of the subvolumes. This free-energy functional is either obtained by analytically integrating out the internal DoFs, {{η}β}, or by computing the free energy by computer simulations for small systems using iterative Boltzmann inversion or force matching.

63.
By the same token, also the translational entropies for a bimodal distribution of the number of internal, positional DoFs per coarse-grained particle are additive in the number of coarse-grained particles.
Fidcg1kBT=β=1nδ1,qβln[ρ^i(rβ)cg1Vi(rβ)/λ3]!ρ^i(rβ)cg1Vi(rβ)/λ3lnVi(rβ)λ3
and
FidcgqkBT=β=1nδq,qβln[ρ^i(rβ)cgqVi(rβ)/λ3]!ρ^i(rβ)cgqVi(rβ)/λ3lnVi(rβ)λ3
.
64.
I.
Pagonabarraga
and
D.
Frenkel
,
J. Chem. Phys.
115
,
5015
(
2001
).
65.
S. Y.
Trofimov
,
E. L. F.
Nies
, and
M. A. J.
Michels
,
J. Chem. Phys.
123
,
144102
(
2005
).
66.
M. S.
Daw
,
S. M.
Foiles
, and
M. I.
Baskes
,
Mater. Sci. Rep.
9
,
251
(
1993
).
67.

The energy of pairwise interactions includes the constant, irrelevant self-interaction term NV(0).

68.
M.
Deserno
,
C.
Holm
, and
S.
May
,
Macromolecules
33
,
199
(
2000
).
69.

The use of a generic three-dimensional cubic grid with linear dimension, ΔL, in conjunction with the approximation of fine discretization for Fcg, requires a judicious choice of ΔL. We illustrate the criteria for choosing ΔL for the case qβq¯: (i) The fine-discretization approximation requires ΔLΔrr0ξ according to Eq. (99). This length scale also allows us to roughly estimate the counterion density in the vicinity of the wire, ρ(r0)ξλ2/(r0ΔL)ξ2λ2/r02. (ii) Representability requires that the number of coarse-grained particles fulfills niρiΔL3/(q¯λ3)O(1); otherwise the typical configuration of the coarse-grained model consists of isolated clusters of q¯ counterions, which is not a typical configuration of the original model. Using the crude estimate of the charge density, we obtain ρiΔL3/(q¯λ3)r0ξq¯λO(1). Thus, using the approximation of fine discretization while simultaneously avoiding representability issues, we require a large inner radius, r0/λ, and a small degree, q¯, of coarse-graining.

70.
L.
Delle Site
and
M.
Praprotnik
,
Phys. Rep.
693
,
1
(
2017
).
71.
A. J.
Rzepiela
,
M.
Louhivuori
,
C.
Peter
, and
S. J.
Marrink
,
Phys. Chem. Chem. Phys.
13
,
10437
(
2011
).
72.
S.
Bevc
,
C.
Junghans
,
K.
Kremer
, and
M.
Praprotnik
,
New J. Phys.
15
,
105007
(
2013
).