Free Energy

Prerequisites: Entropy, Thermodynamics. Full path (9 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

A ball on a hill rolls down until it reaches the lowest point. For a single mechanical object, equilibrium means minimum energy. A system made of very many particles in contact with its surroundings at a fixed temperature behaves differently. The surroundings constantly kick the particles around, and the system ends up in the state that is most likely. A state is likely if it has low energy, but also if it can be realized in many microscopic ways, that is, if it has high entropy.

The free energy combines these two tendencies into one number: energy minus temperature times entropy. At fixed temperature, a system in contact with its surroundings settles into the state of lowest free energy. At low temperature the energy term dominates, and the system chooses ordered states with low energy, like a crystal. At high temperature the entropy term dominates, and the system chooses disordered states that can be realized in many ways, like a gas. Phase transitions such as melting happen where the two contributions balance.

The name comes from a second meaning. When a system changes at constant temperature, not all of its energy can be turned into work, because part of it must flow as heat to keep the entropy bookkeeping right. The decrease of the free energy is the maximum work we can extract. It is the energy that is "free" to be used.

• When Nature plays Skee-ball: the meaning of free energy | Gravity and Levity

Concrete

Definitions. For a system with internal energy $U$, entropy $S$, temperature $T$, pressure $p$ and volume $V$, the Helmholtz and Gibbs free energies are $$F = U - TS, \qquad G = U - TS + pV = F + pV .$$ With the fundamental relation $dU = T\,dS - p\,dV + \mu\, dN$ we get $$dF = -S\,dT - p\,dV + \mu\, dN, \qquad dG = -S\,dT + V\,dp + \mu\, dN .$$ So $F$ is the natural function of $(T, V, N)$ and $G$ of $(T, p, N)$, and for example $S = -(\partial F/\partial T)_{V,N}$ and $p = -(\partial F/\partial V)_{T,N}$.

Minimum principle. Put the system in contact with a heat reservoir at temperature $T$ and keep its volume fixed. If the system absorbs heat $\delta Q$, the reservoir's entropy changes by $-\delta Q/T$, and the second law requires $dS - \delta Q/T \ge 0$. With no work done, $dU = \delta Q$, so $$dU - T\,dS = dF \le 0 \qquad (T, V \text{ fixed}).$$ The free energy can only decrease, and equilibrium is its minimum. At fixed $T$ and $p$ the same argument gives $dG \le 0$, which is why chemists use $G$. If the system does work $W$ on its surroundings at constant $T$, then $W \le -\Delta F$: this is the maximum work statement.

Statistical mechanics. In the canonical ensemble the probability of a microstate $i$ with energy $E_i$ is $p_i = e^{-\beta E_i}/Z$ with $\beta = 1/(k_B T)$ and partition function $Z = \sum_i e^{-\beta E_i}$. The free energy is $$F = -k_B T \ln Z .$$ One checks that this agrees with the thermodynamic definition: $U = \sum_i p_i E_i = -\partial_\beta \ln Z$, and $S = -k_B\sum_i p_i \ln p_i = k_B(\ln Z + \beta U)$, so $U - TS = -k_BT\ln Z$.

Example. A two-level system with energies $0$ and $\varepsilon$ has $Z = 1 + e^{-\beta\varepsilon}$ and $F = -k_BT\ln(1 + e^{-\beta\varepsilon})$. At low temperature $F \to 0$, the energy of the ground state. At high temperature $F \approx -k_BT\ln 2$, which is $-T$ times the entropy $k_B\ln 2$ of two equally occupied states.

Abstract

Legendre transform. In equilibrium thermodynamics $U(S, V, N)$ is a convex function, and the free energy is its Legendre transform in the entropy variable, $$F(T, V, N) = \inf_S \big[\, U(S, V, N) - T S \,\big] .$$ $F$ is therefore concave in $T$, which expresses the positivity of the heat capacity: $\partial^2 F/\partial T^2 = -C_V/T \le 0$. $G$ is obtained by a further Legendre transform in $V$.

Gibbs variational principle. Let $H$ be a Hamiltonian on a finite-dimensional Hilbert space (or a classical system with finitely many states) and define for every density matrix $\rho$ $$F[\rho] = \mathrm{tr}(\rho H) - T\,S[\rho], \qquad S[\rho] = -k_B\, \mathrm{tr}(\rho\ln\rho) .$$ With the Gibbs state $\rho_\beta = e^{-\beta H}/Z$ and $F = -k_BT\ln Z$ one finds $$F[\rho] - F = k_B T\; D(\rho\,\|\,\rho_\beta), \qquad D(\rho\|\sigma) = \mathrm{tr}\,\rho(\ln\rho - \ln\sigma) .$$ The relative entropy $D$ is non-negative and vanishes only for $\rho = \sigma$, so the Gibbs state is the unique minimizer of $F[\rho]$, and the minimum value is the free energy. This is the precise form of the minimum principle, and it is the starting point of mean field theory and of variational methods, which minimize $F[\rho]$ over a restricted family of trial states and obtain upper bounds on $F$.

Thermodynamic limit. For a lattice system in a finite region $\Lambda$, the free energy density $$f(\beta) = -\lim_{\Lambda\to\infty} \frac{1}{\beta\,|\Lambda|}\ln Z_\Lambda$$ exists for a large class of short-range interactions and does not depend on the boundary conditions. For every finite $\Lambda$, $\ln Z_\Lambda$ is analytic in $\beta$ and in the couplings. Non-analyticities can appear only in the limit, and phase transitions are defined as points where $f$ is not analytic. The critical exponents of continuous transitions describe the singular part of $f$ near such points.

Field theory. In quantum field theory at temperature $T$, $Z = \mathrm{tr}\, e^{-\beta H}$ is the Euclidean path integral with time compactified on a circle of circumference $\beta$ (periodic for bosons, antiperiodic for fermions), and $F = -k_BT\ln Z$. The effective action, the Legendre transform of $\ln Z[J]$ with respect to a source $J$, plays the role of a Gibbs free energy as a function of the expectation value of the field.

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Statistics
5. Vector Calculus
6. Entropy
7. Newtonian Mechanics
8. Energy
9. Thermodynamics