We develop a self-consistent field theory for particle dynamics by extremizing the functional integral representation of a microscopic Langevin equation with respect to the collective fields. Although our approach is general, here we formulate it in the context of polymer dynamics to highlight satisfying formal analogies with equilibrium self-consistent field theory. An exact treatment of the dynamics of a single chain in a mean force field emerges naturally via a functional Smoluchowski equation, while the time-dependent monomer density and mean force field are determined self-consistently. As a simple initial demonstration of the theory, leaving an application to polymer dynamics for future work, we examine the dynamics of trapped interacting Brownian particles. For binary particle mixtures, we observe the kinetics of phase separation.

1.
E.
Helfand
,
J. Chem. Phys.
62
,
999
(
1975
).
2.
M. W.
Matsen
and
M.
Schick
,
Phys. Rev. Lett.
72
,
2660
(
1994
).
3.
G. H.
Fredrickson
,
The Equilibrium Theory of Inhomogeneous Polymers
(
Oxford University Press
,
2006
).
4.
R. K. W.
Spencer
and
R. A.
Wickham
,
Soft Matter
9
,
3373
(
2013
).
5.
J. G. E. M.
Fraaije
,
J. Chem. Phys.
99
,
9202
(
1993
).
6.
R.
Hasegawa
and
M.
Doi
,
Macromolecules
30
,
3086
(
1997
).
7.
A. V.
Zvelindovsky
,
G. J. A.
Sevink
,
B. A. C.
van Vlimmeren
,
N. M.
Maurits
, and
J. G. E. M.
Fraaije
,
Phys. Rev. E
57
,
R4879
(
1998
).
8.
T.
Honda
and
T.
Kawakatsu
,
Macromolecules
39
,
2340
(
2006
).
9.
T.
Kawakatsu
,
Phys. Rev. E
56
,
3240
(
1997
).
10.
E.
Reister
,
M.
Müller
, and
K.
Binder
,
Phys. Rev. E
64
,
041804
(
2001
);
E.
Reister
and
M.
Müller
,
J. Chem. Phys.
118
,
8476
(
2003
).
11.
V.
Ganesan
and
V.
Pryamitsyn
,
J. Chem. Phys.
118
,
4345
(
2003
).
12.
M.
Müller
and
G. D.
Smith
,
J. Poly. Sci., Part B: Polym. Phys.
43
,
934
(
2005
).
13.
H.-K.
Janssen
,
Z. Physik B
23
,
377
(
1976
);
R.
Bausch
,
H.-K.
Janssen
, and
H.
Wagner
,
Z. Physik B
24
,
113
(
1976
).
14.
C.
De Dominicis
,
J. Phys. Colloques
37
,
C1
247
(
1976
);
C.
De Dominicis
and
L.
Peliti
,
Phys. Rev. B
18
,
353
(
1978
).
15.
R. V.
Jensen
,
J. Stat. Phys.
25
,
183
(
1981
).
16.
P. C.
Martin
,
E. D.
Siggia
, and
H. A.
Rose
,
Phys. Rev. A
8
,
423
(
1973
).
17.
G. F.
Mazenko
,
Phys. Rev. E
81
,
061102
(
2010
).
18.
S.
Stepanow
,
J. Phys. A: Math. Gen.
17
,
3041
(
1984
).
19.
G. H.
Fredrickson
and
E.
Helfand
,
J. Chem. Phys.
93
,
2048
(
1990
).
20.
V. G.
Rostiashvili
,
M.
Rehkopf
, and
T. A.
Vilgis
,
Eur. Phys. J. B
6
,
497
(
1998
);
V. G.
Rostiashvili
,
M.
Rehkopf
, and
T. A.
Vilgis
,
J. Chem. Phys.
110
,
639
(
1999
).
21.
A.
Crisanti
and
U.
Marini Bettolo Marconi
,
Phys. Rev. E
51
,
4237
(
1995
) have applied a variational principle to an effective MSR action based on Langevin dynamics. Their action is approximate, and is determined through perturbation theory, whereas their variational approach leads to a set of exact dynamical equations for order-parameter averages computed using this approximate action. In our theory, the action is known exactly, and the approximation comes from the application of the variational principle. Our approach leads to an entirely distinct final theoretical framework, involving a self-consistent dynamical mean field.
22.
D. J.
Grzetic
, “
Polymer dynamics: A self-consistent field-theoretic approach
”, Master's thesis,
University of Guelph
,
2011
.
23.
We make the physically reasonable assumption that the initial distribution of chain conformations does not depend on the chain label and drop ℓ from
$P_{0}(\protect \lbrace \protect \bm {R}_{n}(t_{0})\protect \rbrace )$
P0({Rn(t0)})
in Eq. (8).
24.
M.
Doi
and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Oxford University Press
,
1986
).
25.
We note that
$Q([ \psi ],t|t_0) = \intop dx q(x,t|t_0)$
Q([ψ],t|t0)=dxq(x,t|t0)
is a constant, so ρ and q are proportional.
D. S.
Dean
,
J. Phys. A: Math. Gen.
29
,
L613
(
1996
).
27.
U.
Marini Bettolo Marconi
and
P.
Tarazona
,
J. Chem. Phys.
110
,
8032
(
1999
).
28.
G. H.
Fredrickson
and
H.
Orland
,
J. Chem. Phys.
140
,
084902
(
2014
).
You do not currently have access to this content.