(p.230) B Restricted free energy
(p.230) B Restricted free energy
According to statistical mechanics, the free energy of a system at equilibrium can be calculated if the Hamiltonian of the system is known. The Hamiltonian is a function of the set of generalized coordinates $({q}_{1},{q}_{2},\dots ,{q}_{f})$ and momenta $({p}_{1},{p}_{2},\dots ,{p}_{f})$. For simplicity we shall denote the $2f$ set of variables by Γ,
Then the Hamiltonian is written as
Given $H(\mathrm{\Gamma})$, the free energy of the system at temperature T is calculated by
where $\mathrm{\beta}=1/{k}_{B}T$.
Equation ( B.3 ) represents the free energy of the system which is at true equilibrium under a given temperature T.
In thermodynamics, we often consider the free energy of a system which is at equilibrium under some hypothetical constraints. Such a free energy is called a restricted free energy. For example, in discussing the phase separation of solutions, we consider the system consisting of two homogeneous solutions (each having concentration ${\varphi}_{1}$, ${\varphi}_{2}$ and volume ${V}_{1}$, ${V}_{2}$), and then minimize the total free energy ${F}_{tot}({\varphi}_{1},{\varphi}_{2},{V}_{1},{V}_{2};T)$ with respect to ${\varphi}_{i}$ and ${V}_{i}$ to obtain the true equilibrium state. The free energy ${F}_{tot}({\varphi}_{1},{\varphi}_{2},{V}_{1},{V}_{2};T)$ is an example of the restricted free energy.
The restricted free energy appears many times in this book. For example, in Section 3.2.3 , we considered the free energy $U(r)$ of a polymer chain whose endtoend vector is fixed at $r$. In Section 5.2.3 , we considered the free energy $F(S;T)$ of a liquid crystalline system in which the order parameter is fixed at S. In Section 7.2.1 , we considered the free energy $A(x)$ for the nonequilibrium state specified by variable x. In this appendix, we discuss the restricted free energy from the general viewpoint of statistical mechanics.
(p.231) Let us consider a system in which the values of certain physical quantities ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$ ($i=1,2,\dots ,n$) are fixed at ${x}_{i}$, i.e.,
The free energy of such a system is given by
where x stands for $({x}_{1},{x}_{2},\dots ,{x}_{n})$. The constraints can be accounted for by introducing the hypothetical potential
where ${k}_{i}$ is a large positive constant which gives a penalty for the deviation of the quantity ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$ from the given value ${x}_{i}$. Using the formula
the free energy of the constrained system can be written as
where the coefficient ${\mathrm{\Pi}}_{i}({k}_{i}\mathrm{\beta}/2\mathrm{\pi}{)}^{1/2}$ has been omitted since it only gives terms independent of ${x}_{i}$.
From eq. ( B.5 ), it is easy to derive the following properties of the restricted free energy.

(1) If the system is at equilibrium without the constraints, the probability that the physical quantity ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$ has the value ${x}_{i}$ is given by
(B.9)$$P(x)\propto \text{}{e}^{\mathrm{\beta}A(x,T)}$$ 
(2) The free energy of the system at equilibrium is given by
(B.10)$$A(T)=\frac{1}{\mathrm{\beta}}ln\left[\int dx\text{}{e}^{\mathrm{\beta}A(x,T)}\right]$$
Equations ( B.9 ) and ( B.10 ) indicate that $A(x,T)$ is playing the role of the Hamiltonian $H(\mathrm{\Gamma})$ in the new phase space specified by $x=({x}_{1},{x}_{2},\dots ,{x}_{n})$.
According to eq. ( B.9 ), the most probable state is the state x which minimizes $A(x,T)$. This corresponds to the thermodynamic principle that the true equilibrium state is the state which minimizes the restricted free energy $A(x,T)$.
(p.232) To fix the values of the physical quantities at x, external forces need to be applied to the system. For example, to keep the endtoend vector of a polymer chain at the fixed value $r$, a force $\stackrel{\u02c6}{f}$ needs to be applied at the chain ends (see the discussion in Section 3.2.3 ). The force needed to keep ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$ at ${x}_{i}$ is given by
The average of ${\stackrel{\u02c6}{F}}_{i}(\mathrm{\Gamma},x)$ is given by
This is the generalization of eq. ( 3.30 ).
The restricted free energy $A(x,T)$ can be calculated by eqs. ( B.5 ) or ( B.8 ). There is another way of calculating $A(x,T)$. This method has already been used in Section 3.2.3 to calculate the restricted free energy $U(r)$ of a polymer chain whose endtoend vector is fixed at $r$. There, instead of fixing the endtoend vector at $r$, we assumed that a constant force $f$ is applied at the chain end. If $f$ is suitably chosen, the average endtoend vector becomes equal to the given value $r$. ^{1} If the relation between $f$ and $r$ is known, the restricted free energy $U(r)$ is obtained by eq. ( 3.30 ).
The above method can be generalized as follows. We consider the Hamiltonian
where ${h}_{i}$ represents the hypothetical external force conjugate to ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$. We calculate the free energy $\tilde{A}({h}_{i},T)$ for the Hamiltonian ( B.13 )
The average of ${\stackrel{\u02c6}{x}}_{i}(\mathrm{\Gamma})$ for given forces $({h}_{1},{h}_{2},\dots )$ is calculated as
(p.233) On the other hand, the free energy $\tilde{A}(h,T)$ is equal to the minimum of $A(x,T)\sum _{i}{h}_{i}{x}_{i}$ with respect to ${x}_{i}$. Therefore
Hence
and $A(x,T)$ is written as
Equations ( B.16 ) and ( B.15 ) are the Legendre transform. Therefore, the restricted free energy $A(x,T)$ is obtained either by eq. ( B.16 ), or by ( B.15 ). We now consider two examples of the restricted free energy.
First we consider a twocomponent solution. Let ${r}_{pi}$ ($i=1,2,\dots ,{N}_{p})$ and ${r}_{si}$ ($i=1,2,\dots ,{N}_{s})$ be the position vectors of solute molecules and solvent molecules. The potential energy of the system is written as
where ${u}_{pp}(r)$, ${u}_{ps}(r)$, and ${u}_{ss}(r)$ are the potentials for solute–solute, solute–solvent, and solvent–solvent interactions.
The effective interaction potential between solute molecules is given by the restricted free energy:
This represents the effective interaction between the solute molecules with the effect of the solvent included. The effective interaction $\mathrm{\Delta}\u03f5$ introduced in eq. ( 2.49 ) is an example of such a potential for the lattice model. The interaction potential $U(h)$ between the colloidal particles introduced in Section 2.5.1 is another example of such a potential.
In the theory of solutions, ${A}_{pp}({r}_{p})$ is called the potential of the mean force since the average of the force acting on solute i is given by the partial derivative of ${A}_{pp}({r}_{p})$:
This is a special case of eq. ( B.10 ). (The minus sign arises from the difference in the definition of the force.) Notice that even though the original potential ${U}_{tot}$ is written as a sum of twobody potentials such (p.234) as ${u}_{pp}(r)$, ${u}_{ps}(r)$, and ${u}_{ss}(r)$, the potential ${A}_{pp}$ is generally not written as a sum of the twobody potentials.
As the next example, let us consider the free energy introduced in Section 5.2.3 .
We consider N nematic forming molecules, each pointing along the unit vector ${u}_{i}$ ($i=1,2,\dots ,N$). We assume that the scalar order parameter of the system is S, i.e., ${u}_{i}$ satisfy the constraint of eq. ( 5.28 ). To take into account this constraint, we assume that the Hamiltonian of the system is written as
where we have assumed (for simplicity) that the molecules are not interacting with each other (i.e., $U=0$).
The free energy $\tilde{A}(h)$ can be written as
where $\tilde{A}(0)$ is the free energy of the system with no constraint, and $\u27e8\cdots {\u27e9}_{0}$ is the average for $u$ for an isotropic distribution of molecules. In the following, we set $\tilde{A}(0)$ equal to zero.
The righthand side of eq. ( B.21 ) can be calculated as a power series with respect to h:
Hence
Therefore the order parameter S of the system is given by
Hence
(p.235) If there is an interaction between the particles, the interaction energy is written as (see eq. ( 5.11 ))
Hence the free energy is given by
where ${T}_{c}=2U/15{k}_{B}$. Equation ( B.29 ) is the first term in the expansion of eq. ( 5.35 ).