Overview of ensembles#

  1. Entropy as a function of microstate probabilities:

    \[S = -k \sum_{i} p_i \log p_i\]

  2. Constraint imposed on thermodynamic coordinates to maintain constant values:

    • Normalization of probabilities(all ensembles):

      \[\sum_{i} p_i = 1\]

    • Expected energy (for certain ensembles):

      \[\sum_{i} p_i E_i = \langle E \rangle\]

    • Expected number of particles (for certain ensembles):

      \[\sum_{N} p_N N = \langle N \rangle\]

    • Expected volume (for certain ensembles):

      \[\sum_{i} p_V V = \langle V \rangle\]

  3. Lagrange Multipliers: Construct the Lagrangian with Lagrange multipliers} \(\alpha\), \(\beta\), …

    \[L = -k \sum_{i} p_i \log p_i - \alpha \left( \sum_{i} p_i - 1 \right) - \beta \left( \sum_{i} p_i E_i - ...\]

Finding maximum entropy solution with constraints (NVE example)#

Seek maximum of entropy

\[\frac{\partial L}{\partial p_j} = -k_B (\log p_j + 1) - \alpha = 0\]
\[p_j = e^{-\frac{\alpha}{k_B } - 1}\]

Determine Lagrange multipliers

  • Normalize \(p_j\) to find \(\alpha\):

    \[\sum_j p_j = \sum_j e^{-\frac{\alpha}{k_B} - 1} =\Omega\]

  • probabilities are independent of the microstate, and the sum can be set to be equal to a constant denoted as \(\Omega\)

Probability Expression:

\[ p_j = \frac{1}{\Omega}\]

Finding maximum entropy solution with constraints (NVT example)#

\[\frac{\partial L}{\partial p_j} = -k (\log p_j + 1) - \alpha - \beta E_j = 0\]
\[p_j = e^{-\frac{\alpha}{k} - 1} e^{-\frac{\beta E_j}{k}}\]

Determine Lagrange multipliers

  • Normalize \(p_j\) to find \(\alpha\):

    \[e^{-\frac{\alpha}{k} - 1} = \frac{1}{Z}\]
    \[Z = \sum_{j} e^{-\frac{\beta E_j}{k}}\]

  • \(\beta\) is typically identified with \(\frac{1}{kT}\).

Probability Expression:

\[ p_j = \frac{e^{-\frac{\beta E_j}{k}}}{Z}\]

NVE Overview#

Thermodynamics#

  • Fundamental relation

\[dE = TdS-pdV+\mu dN\]

  • Derivatives of energy

\[T = \Big(\frac{\partial E}{\partial S}\Big)_{V,N}\,\,\,\,\,\, p = -\Big(\frac{\partial E}{\partial V}\Big)_{S,N} \,\,\,\,\,\, \mu = \Big(\frac{\partial E}{\partial N}\Big)_{S,V}\]

  • Second law

\[dS(E,V,N)\geq 0\,\,\,\,\,\,\,\,\, dE(S,V,N)\leq 0\]

Statistical mechanics#

  • Bridge equation

\[\Omega(N,V,E) = \sum^{}_{i} 1\]
\[S(N,V,E) = k_B log \Omega(N, V, E)\]

  • Probability of a microstate

\[p(E) = \frac{1}{\Omega(N, V, E)}\]

  • Probability of a macrostate

\[p(E_1, E-E_1) = \frac{\Omega(E_1) \Omega(E-E_1)}{\sum_{E_1} \Omega(E_1) \Omega(E-E_1)} = \frac{e^{S_1(E_1)+S_2(E-E_1)}}{e^{S(E)}}\]

NVT Overview#

Thermodynamics#

  • Fundamental relation

\[dF = d(E-TS) = -SdT -pdV + \mu dN\]

  • Derivatives of free energy

\[S = -\Big(\frac{\partial F}{\partial T}\Big)_{V,N}\,\,\,\,\,\, p = -\Big(\frac{\partial F}{\partial V}\Big)_{T,N} \,\,\,\,\,\, \mu = \Big(\frac{\partial F}{\partial N}\Big)_{T,V}\]

  • Second law

\[dF(T,V,N)\leq 0\]

Statistical mechanics#

  • Bridge equation

\[Z(N,V,T) = \sum^{}_{i} e^{-E_i/k_B T} = \sum_E \Omega(E) e^{-E/k_B T}\]
\[F=-k_BT log Z\]

  • Probability of a microstate

\[p(E_i) = \frac{e^{-E_i/k_BT}}{Z}\]

  • Probability of a macrostate

\[p(E') = \frac{\Omega(E^{'}) e^{-E^{'}/k_BT}}{Z} = \frac{e^{-\beta F^{'}}}{Z}\]

\(\mu\)VT Overview#

Thermodynamics#

  • Fundamental relation

\[d\Phi_{\mu, T} = d(U-TS - \mu N) = -SdT-pdV-Nd\mu\]

  • Derivatives of free energy

\[S = -\Big(\frac{\partial \Phi}{\partial T}\Big)_{V,N}\,\,\,\,\,\, p = -\Big(\frac{\partial \Phi}{\partial V}\Big)_{T,\mu} \,\,\,\,\,\, N = - \Big(\frac{\partial \Phi}{\partial \mu}\Big)_{T,V}\]

  • Second law

\[d\Phi\leq 0\]

Statistical mechanics#

  • Bridge equation

\[Z_G(\mu,V,T) = \sum^{}_{i, N} e^{-E_i/k_B T} \cdot e^{\mu N/k_B T} = \sum_E \Omega(E, N) e^{-E/k_B T} e^{\mu N/k_B T}\]
\[\Phi(\mu, V, T) =-k_BT log Z_G\]

  • Probability of a microstate

\[p(E_i) = \frac{e^{-E_i/k_BT} e^{\mu N_i/k_BT}}{Z_G}\]

  • Probability of a macrostate

\[p(E', N') = \frac{\Omega(E', N') e^{-E'/k_BT} e^{ \beta \mu N'}}{Z_G} = \frac{e^{-\beta \Phi'}}{Z_G}\]

Ensemble equivalence#

Smallness of fluctuations#

\[dU = SdT -VdP +\mu dN + BdM +... = \sum_i Y_i dX_i\]

Consider a conjucate pair of extensive \(X_i\) and intensive \(Y_i\) variables. For instnace \((1, E)\), \((S, T)\) or \((V, -p)\). The average and fluctuations of extensive variable \(X\) for constant \(Y\) is given by derivatives of parition function:

\[\langle X \rangle = \frac{\partial log Z}{\partial \beta Y}\]
\[\sigma^2_X = \langle X^2 \rangle - \langle X \rangle^2 = \frac{\partial^2 log Z}{\partial (\beta Y)^2}\]

  • Smallness of fluctuations

\[\sigma^2_E = k_B T^2 C_v(T) \]
\[\sigma^2_N = \frac{k_B T}{v} \kappa_T\]

Legendre and Laplace transforms#

Legendre transformation enables expressing the condition of equilibrium (e.g. maximium of entropy function) in terms of convenient variables (e.g minimum of some free eneergy function).

  • Free energies are Legedre transforms of internal energy function \(E(S,V,N,...)\)

\[\mathcal{L}_{S} E(S,V,N) = U - T\cdot S = F(N, V, T)\]
\[\mathcal{L}_{S, V} E(S,V,N) = U - T\cdot S + pV = G(N, p, T)\]

General expression of Legendtre transform

A general expression for legendre transform of energy \(U(X_0, X_1, ...X_n, X_n+1, ... X_t)\) with respect to its extensive variables can be written down as:

\[\mathcal{L}_{X_{0}, ... X_{n}} U = U - \sum Y_k X_k = \Psi(Y_0,... Y_{n}, X_{n+1}, ...X_{t})\]

The various partion functions can then be seen to be of general form \(e^{\beta \Psi}\) where the free energy function is a Legendre transform over fluctuating quantities.

\[Z(X_0, ... X_n | X_{n+1}, ... X_{t}) = exp \big(-\beta \mathcal{L_{X_{0}, ... X_{n}}} E (X_0, ... X_t) \big)\]

  • Free energies as Laplace transform of internal energy function \(E(S,V,N,...)\)

\[Z(\beta, N, V) = e^{-\beta F(\beta, N, V)}\]