Energy and entropy are two of the most important concepts in science. For all natural processes where a system exchanges energy with its environment, the energy of the system tends to decrease and the entropy of the system tends to increase. Free energy is the special concept that specifies how to balance the opposing tendencies to minimize energy and maximize entropy. There are many pedagogical articles on energy and entropy. Here we present a simple model to illustrate the concept of free energy and the principle of minimum free energy.
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Why does a non-isolated system tend to lower its mechanical energy? The resulting transfer of energy from the system to the environment of temperature T brings about an increase in the entropy of the environment. An environment of low T can gain a large amount of entropy since entropy gain = (energy gain by heating)/T. If the entropy gain of the environment outweighs the entropy lost by the system, then the entropy of the universe will increase and the process can naturally occur according to the second law of thermodynamics.
If the “ball” is a golf ball instead of a nitrogen molecule, then the thermal interaction between the gas and the reservoir can be mechanically approximated by vibrating the bottom wall of the container. When the ball (gas particle) collides with this wall oscillator (solid atom), the ball gains or loses energy so as to maintain an equilibrium (Boltzmann-like) distribution of speeds. Heating the gas in this mechanical model can be accomplished by increasing the vibration amplitude of the bottom wall.
By definition, W counts the number of microstates with a certain E. If E and H are known precisely, then the ball simply bounces up and down at constant speed v in the space of length H. Gravity has little effect on changing the speed due to the high-speed condition, v2>> gH, or equivalently kT >> mgH. Thus, for fixed E and H, the ball spends the same amount of time in each accessible cell and in each possible velocity state (+v and −v). Each microstate is equally probable.
The result S = k ln H + S0 can also be derived without cells—by treating position x and velocity v as continuous variables. If the energy is known to lie between E and E + δE due to experimental uncertainty, then the possible microstates (x, v) lie in a certain area of phase space. In classical statistical mechanics, W is proportional to this area: W ∝ ∬dxdv = 2Hδv, and thus k ln W = k ln H + constant. For both the geometric (phase space) and the combinatoric (cell model) calculations, the fact that W is multiplicative, W = H × Wvel, implies that S is additive: S = k ln H + k ln Wvel.
Ref. 11, pp. 511–512. In terms of temperature (T = mv2/k) and entropy (S = k ln H), we can write the molecular force formula, f = mv2/H, in the form f = TdS/dH.
The amplitude of the piston oscillation is ½g(t/2)2 = (m/2M)He, which is a negligible fraction of He for m >>M. Note that these small mechanical (periodic) fluctuations in the height of the piston occur under special conditions where v is exactly constant (no reservoir) and resonance is exactly satisfied (piston period matches ball period). For the one-ball system at constant T, the thermal fluctuations in the height of the piston are random and not negligible because there exists a distribution of v's controlled by a reservoir.
The number of microstates (spatial locations) accessible to one molecule moving inside a volume V is proportional to V (imagine subdividing the volume into unit cells). The number of microstates accessible to N independent molecules is W − VN. Since V ∝ H, we have W ∝ HN. The entropy S = k ln W becomes S = k ln HN + constant.