Physical Constraints for Equilibrium#

For a system to maintain equilibrium values, the microstate probabilities must satisfy the following conditions:

1. Normalization:

   $$\sum _i{p}_i=1$$

2. Constraint to Maintain the Expectation Value of an Observable $X$ (e.g., Energy or Volume):

   $$\sum _i{p}_i{X}_i=⟨X⟩,\sum _i{p}_i{Y}_i=⟨Y⟩$$

• The probability of a macrostate $(X,Y,\dots )$ follows the general form:

  $$P(X,Y,\dots )=\frac{e^{\frac{1}{k_B}S(X,Y,..)}\cdot e^{-\beta X}\cdot e^{-\gamma Y}...}{Z}$$
  $$Z(\beta ,\gamma ,...)=\sum _{X,Y,...}{e}^{\frac{1}{k_B}S(X,Y,..)}{e}^{-\beta X}{e}^{-\gamma Y}...$$

• Where $Z$ is a normalization factor called partition function

• Exponential dependence can also be seen as consequence of exchange of (energy, volume ,etc) with an environment or a large reserovoir.

Comparison of Ensembles#

• Entropy dependence $e^{S/k_B}$ is universal across all ensembles.

• Microstate probability follows different forms based on constraints from different thermodynamic potentials.

• Macrostate probability always includes an entropy term but is modified by energy, pressure, and chemical potential, depending on the ensemble.

Extensive vs intensive variables#

• The total differential of internal energy $U$ in a thermodynamic system can be expressed in terms of its conjugate variables:

• where each pair $(X_i,f_i)$ represents a conjugate extensive and intensive variable respectively, such as:

  • $(S,T)$ → entropy and temperature,

  • $(V,-P)$ → volume and pressure,

  • $(N,\mu )$ → particle number and chemical potential,

  • $(M,B)$ → magnetization and magnetic field.

Laplace Transform and Ensemble Connections#

• The Laplace transform connects different thermodynamic ensembles by linking the partition function to energy and volume fluctuations. It effectively approximates a Legendre transform in the thermodynamic limit.

• Canonical Ensemble (Energy Integration):

  $$Z(\beta ,N,V)=\int dE\mathrm{\Omega }(E)e^{-\beta E}\approx e^{\underset{E}{min}[S(E)/k_B-\beta E]}$$
  $$Z(\beta ,N,V)=e^{-\beta [U-TS]}=e^{-\beta F(\beta ,N,V)}$$

• Isothermal-Isobaric Ensemble (Volume Integration):

  $$Z(\beta ,N,P)=\int dVZ(\beta ,N,V)e^{-\beta PV}\approx e^{\underset{V}{min}[F(V)-\beta PV]}$$
  $$Z(\beta ,N,P)=e^{-\beta [U-TS-PV]}=e^{-\beta G(\beta ,N,P)}$$

• Thus, free energy functions naturally emerge as Legendre transforms of internal energy through Laplace integration over fluctuating variables.

Legendre Transform and Thermodynamic Potentials#

• The Legendre transformation allows us to reformulate equilibrium conditions (e.g., entropy maximization) in terms of more convenient variables (e.g., free energy minimization).

• This transformation makes it possible to work with temperature and pressure as control variables instead of entropy and volume.

• Free Energies as Legendre Transforms of Internal Energy

• Helmholtz Free Energy (Legendre transform over $S$):

• Gibbs Free Energy (Legendre transform over $S,V$):

Fluctuation-Response Relations#

• For a given extensive variable $X$ and its conjugate intensive variable $f$, the partition function $Z$ governs both the mean value and fluctuations of $X$.

• Mean value of $X$ at constant $f$:

  $$⟨X⟩=\frac{\partial \log Z}{\partial (\beta f)}$$

• Fluctuations of $X$ at constant $Y$:

  $${\sigma }_X^2=⟨X^2⟩-⟨X{⟩}^2=\frac{{\partial }^2\log Z}{\partial (\beta f{)}^2}$$

• This relation shows that fluctuations in $X$ are directly linked to the second derivative of the partition function, a fundamental result of statistical mechanics.