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PHY452H1S Basic Statistical Mechanics. Lecture 11: Statistical and thermodynamic connection. Taught by Prof.\ Arun Paramekanti

Posted by peeterjoot on February 27, 2013

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Peeter’s lecture notes from class. May not be entirely coherent.

Connections between statistical and thermodynamic views

  • “Heat”. Disorganized energy.
  • S_{\text{Statistical entropy}}. This is the thermodynamic entropy introduced by Boltzmann (microscopic).

Ideal gas

\begin{aligned}H = \sum_{i = 1}^N \frac{\mathbf{p}_i^2}{2m}\end{aligned} \hspace{\stretch{1}}(1.3.1)

\begin{aligned}\Omega(E) = \frac{1}{{h^{3N} N!}}\int d\mathbf{x}_1 d\mathbf{x}_2  \cdots  d\mathbf{x}_Nd\mathbf{p}_1 d\mathbf{p}_2  \cdots  d\mathbf{p}_N\delta( E - H )\end{aligned} \hspace{\stretch{1}}(1.3.2)

Let’s isolate the contribution of the Hamiltonian from a single particle and all the rest

\begin{aligned}H = \frac{\mathbf{p}_1^2}{2m}+\sum_{i \ne 1}^N \frac{\mathbf{p}_i^2}{2m}=\frac{\mathbf{p}_1^2}{2m}+H'\end{aligned} \hspace{\stretch{1}}(1.3.3)

so that the number of states in the phase space volume in the phase space region associated with the energy is

\begin{aligned}\Omega(N, E) &= \frac{V^N}{h^{3N} N!}\int d\mathbf{p}_1\int d\mathbf{p}_2 d\mathbf{p}_3 \cdots d\mathbf{p}_N\delta( E - H' - H_1) \\ &= \frac{V^{N-1}}{h^{3(N-1)} (N-1)!} \frac{V}{h^3 N}\int d\mathbf{p}_1\int d\mathbf{p}_2 d\mathbf{p}_3 \cdots d\mathbf{p}_N\delta( E - H' - H_1) \\ &= \frac{ V }{ h^3 N} \int d\mathbf{p}_1 \Omega( N-1, E - H_1 )\end{aligned} \hspace{\stretch{1}}(1.3.4)

With entropy defined by

\begin{aligned}S = k_{\mathrm{B}} \ln \Omega,\end{aligned} \hspace{\stretch{1}}(1.3.5)

we have

\begin{aligned}\Omega( N-1, E - H_1 ) = \exp\left(  \frac{1}{k_{\mathrm{B}}} S \left( N-1, E - \frac{\mathbf{p}_1^2}{2m} \right)  \right)\end{aligned} \hspace{\stretch{1}}(1.3.6)

so that

\begin{aligned}\Omega(N, E) =\frac{ V }{ h^3 N} \int d\mathbf{p}_1 \exp\left(  \frac{1}{k_{\mathrm{B}}} S \left( N-1, E - \frac{\mathbf{p}_1^2}{2m} \right)  \right)\end{aligned} \hspace{\stretch{1}}(1.3.7)

For N \gg 1 and E \gg \mathbf{p}_1^2/2m, the exponential can be approximated by

\begin{aligned}\exp\left(  \frac{1}{k_{\mathrm{B}}} S \left( N-1, E - \frac{\mathbf{p}_1^2}{2m} \right)  \right)= \exp\left(  \frac{1}{k_{\mathrm{B}}} \left(  S(N, E) - \left( \frac{\partial {S}}{\partial {N}} \right)_{E, V} - \frac{\mathbf{p}_1^2}{2m} \left( \frac{\partial {S}}{\partial {E}} \right)_{N, V}  \right)  \right),\end{aligned} \hspace{\stretch{1}}(1.3.8)

so that

\begin{aligned}\Omega(N, E) = \underbrace{\frac{ V }{ h^3 N} \int d\mathbf{p}_1 e^{\frac{S}{k_{\mathrm{B}}}(N, E)}e^{-\frac{1}{{k_{\mathrm{B}}}}\left( \frac{\partial {S}}{\partial {N}} \right)_{E, V}}}_{B}\int d\mathbf{p}_1 e^{-\frac{\mathbf{p}_1^2}{2m k_{\mathrm{B}}}\left( \frac{\partial {S}}{\partial {E}} \right)_{N, V}}.\end{aligned} \hspace{\stretch{1}}(1.3.9)


\begin{aligned}\Omega(N, E) = B\int d\mathbf{p}_1 e^{-\frac{\mathbf{p}_1^2}{2m k_{\mathrm{B}}}\left( \frac{\partial {S}}{\partial {E}} \right)_{N, V}}.\end{aligned} \hspace{\stretch{1}}(1.3.10)

\begin{aligned}\mathcal{P}(\mathbf{p}_1) \propto e^{-\frac{\mathbf{p}_1^2}{2m k_{\mathrm{B}} T}}.\end{aligned} \hspace{\stretch{1}}(1.3.11)

This is the Maxwell distribution.

Non-ideal gas. General classical system


Fig 1: Partitioning out a subset of a larger system

Breaking the system into a subsystem 1 and the reservoir 2 so that with

\begin{aligned}H = H_1 + H_2\end{aligned} \hspace{\stretch{1}}(1.4.12)

we have

\begin{aligned}\Omega(N, V, E) &= \int d\{x_1\}d\{p_1\}d\{x_2\}d\{p_2\}\delta( E - H_1 - H_2 ) \frac{1}{{ h^{3N_1} N_1! h^{3 N_2} N_2!}} \\ &\propto \int d\{x_1\}d\{p_1\}e^{\frac{1}{{k_{\mathrm{B}}}} S(E - H_1, N - N_1)}\end{aligned} \hspace{\stretch{1}}(1.4.13)

\begin{aligned}\Omega(N, V, E) \sim \int d\{x_1\}d\{p_1\}\underbrace{e^{\frac{1}{{k_{\mathrm{B}}}}S(E, N)}e^{-\frac{N_1 }{k_{\mathrm{B}}}\left(  \frac{\partial {S}}{\partial {N}}  \right)_{E, V}}}_{\text{``environment'', or ``heat bath''}}e^{-\frac{H_1 }{k_{\mathrm{B}}}\left(  \frac{\partial {S}}{\partial {E}}  \right)_{N, V}}\end{aligned} \hspace{\stretch{1}}(1.4.14)

\begin{aligned}H_1 = \sum_{i \in 1} \frac{\mathbf{p}_i}{2m}+\sum_{i \in j} V(\mathbf{x}_i - \mathbf{x}_j)+ \sum_{i \in 1} \Phi(\mathbf{x}_i)\end{aligned} \hspace{\stretch{1}}(1.4.15)

\begin{aligned}\mathcal{P}  \propto e^{-\frac{H( \{x_1\} \{p_1\} ) }{k_{\mathrm{B}} T} }\end{aligned} \hspace{\stretch{1}}(1.4.16)

and for the subsystem

\begin{aligned}\mathcal{P}_1 =\frac{e^{-\frac{H_1}{k_{\mathrm{B}} T} }}{\int d\{x_1\}d\{p_1\}e^{-\frac{H_1}{k_{\mathrm{B}} T} }}\end{aligned} \hspace{\stretch{1}}(1.4.17)

Canonical ensemble

Can we use results for this subvolume, can we use this to infer results for the entire system? Suppose we break the system into a number of smaller subsystems as in fig. 1.2.


Fig 2: Larger system partitioned into many small subsystems

\begin{aligned}\underbrace{(N, V, E)}_{\text{microcanonical}}\rightarrow (N, V, T)\end{aligned} \hspace{\stretch{1}}(1.5.18)

We’d have to understand how large the differences between the energy fluctuations of the different subsystems are. We’ve already assumed that we have minimal long range interactions since we’ve treated the subsystem 1 above in isolation. With \beta = 1/(k_{\mathrm{B}} T) the average energy is

\begin{aligned}\left\langle{{E}}\right\rangle = \frac{\int d\{x_1\}d\{p_1\}He^{- \beta H }}{\int d\{x_1\}d\{p_1\}e^{- \beta H }}\end{aligned} \hspace{\stretch{1}}(1.5.19)

\begin{aligned}\left\langle{{E^2}}\right\rangle = \frac{\int d\{x_1\}d\{p_1\}H^2e^{- \beta H }}{\int d\{x_1\}d\{p_1\}e^{- \beta H }}\end{aligned} \hspace{\stretch{1}}(1.5.20)

We define the partition function

\begin{aligned}Z \equiv \frac{1}{{h^{3N} N!}}\int d\{x_1\}d\{p_1\}e^{- \beta H }.\end{aligned} \hspace{\stretch{1}}(1.5.21)

Observe that the derivative of Z is

\begin{aligned}\frac{\partial {Z}}{\partial {\beta}} = -\frac{1}{{h^{3N} N!}}\int d\{x_1\}d\{p_1\}He^{- \beta H },\end{aligned} \hspace{\stretch{1}}(1.5.22)

allowing us to express the average energy compactly in terms of the partition function

\begin{aligned}\left\langle{{E}}\right\rangle = -\frac{1}{{Z}} \frac{\partial {Z}}{\partial {\beta}} = - \frac{\partial {\ln Z}}{\partial {\beta}}.\end{aligned} \hspace{\stretch{1}}(1.5.23)

Taking second derivatives we find the variance of the energy

\begin{aligned}\frac{\partial^2 {{\ln Z}}}{\partial {{\beta}}^2} &=\frac{\partial {}}{\partial {\beta}}\frac{\int d\{x_1\}d\{p_1\}(-H)e^{- \beta H }}{\int d\{x_1\}d\{p_1\}e^{- \beta H }} \\ &= \frac{\int d\{x_1\}d\{p_1\}(-H)^2e^{- \beta H }}{\int d\{x_1\}d\{p_1\}e^{- \beta H }}-\frac{\left(  \int d\{x_1\} d\{p_1\} (-H) e^{- \beta H }  \right)^2}{\left(  \int d\{x_1\} d\{p_1\} e^{- \beta H }  \right)^2} \\ &= \left\langle{{E^2}}\right\rangle - \left\langle{{E}}\right\rangle^2 \\ &= \sigma_{\mathrm{E}}^2\end{aligned} \hspace{\stretch{1}}(1.5.24)

We also have

\begin{aligned}\sigma_{\mathrm{E}}^2 &= -\frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {\beta}} \\ &= \frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {T}} \frac{\partial {T}}{\partial {\beta}} \\ &= -\frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {T}} \frac{\partial {}}{\partial {\beta}} \frac{1}{{k_{\mathrm{B}} \beta}} \\ &= \frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {T}} \frac{1}{{k_{\mathrm{B}} \beta^2}} \\ &= k_{\mathrm{B}} T^2 \frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {T}}\end{aligned} \hspace{\stretch{1}}(1.5.25)

Recalling that the heat capacity was defined by

\begin{aligned}C_V = \frac{\partial {\left\langle{{E}}\right\rangle}}{\partial {T}},\end{aligned} \hspace{\stretch{1}}(1.5.26)

we have

\begin{aligned}\sigma_{\mathrm{E}}^2 = k_{\mathrm{B}} T^2 C_V  \propto  N\end{aligned} \hspace{\stretch{1}}(1.5.27)

\begin{aligned}\frac{\sigma_{\mathrm{E}}}{\left\langle{{E}}\right\rangle}  \propto  \frac{1}{{\sqrt{N}}}\end{aligned} \hspace{\stretch{1}}(1.5.28)


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