Thermodyanmics Properties

3.2. Thermodyanmics Properties#

3.2.1. Classical Systems#

Classical systems have continuous positions and momenta, so sums become integrals:

\[ Q(T, V, N) = \frac{1}{N! h^{3N}} \int d^{3N}q \, d^{3N}p \, e^{-\beta H(q, p)} \]

  • Indistinguishable particles

  • Prefactor to account for quantum entropy in high T limit where particles are exchanged and are the same.

For classical systems, ( H(q, p) = U(q) + K(p) ) is separable:

  • ( U(q) ): potential energy

  • ( K(p) = \sum \frac{p^2}{2m} ): kinetic energy

Hence, ( Q ) is separable:

\[ Q(T, V, N) = \frac{1}{N! h^{3N}} \int d^{3N}q \, e^{-\beta U(q)} \int d^{3N}p \, e^{-\beta K(p)} \]

  • Configuration integral:

\[ Z(T, V, N) = \int d^{3N}q \, e^{-\beta U(q)} \]

  • Thermal de Broglie wavelength:

\[ \Lambda = \left( \frac{h^2}{2\pi m k_B T} \right)^{1/2} \]

Example: Ideal gas ( U(q) = 0 ), so

\[ Z = V^N, \quad Q = \frac{V^N}{N! \Lambda^{3N}} \]

Free energy:

\[ \beta A = -\ln Q = -\ln \left( \frac{V^N}{N! \Lambda^{3N}} \right) = N \ln \left( \frac{\Lambda^3}{V} \right) + N \]

3.2.2. Probability Distributions#

Probability density functions are also separable:

\[ f(q, p) = \frac{1}{Q} e^{-\beta H(q, p)} = \frac{1}{Z} e^{-\beta U(q)} \cdot \frac{1}{K} e^{-\beta K(p)} \]

  • Marginal distributions:

    • Maxwell-Boltzmann distribution:

\[ f(p_i) = \left( \frac{\beta}{2\pi m} \right)^{1/2} e^{-\beta p_i^2 / 2m} \]

Each momentum component is independently Gaussian distributed. The variance increases with temperature.

3.2.3. Temperature#

Using the distribution of ( p ), we can find the average kinetic energy:

\[ \langle K \rangle = \int d^{3N}p \, K(p) f(p) = \frac{3N}{2} k_B T \]

In the canonical ensemble, ( T ) is fixed but ( K ) fluctuates. We define a “kinetic” temperature:

\[ T = \frac{2 \langle K \rangle}{3N k_B} \]

In practice, some momenta are constrained. For example, if linear momentum is conserved, only ( N - 1 ) momenta are true variables. So we define:

\[ T = \frac{2 \langle K \rangle}{N_{\text{DOF}} k_B} \]

Most common form: ( N_{\text{DOF}} = 3(N - 1) ) for linear momentum conservation.

3.2.4. Pressure#

The pressure can be computed from ( Q ):

\[ \beta P = \left( \frac{\partial \ln Q}{\partial V} \right)_{T, N} \]

Be careful: the integration limits of ( Q ) depend on ( V )!

Scale the coordinates as ( q’ = q / V^{1/3} ), then:

\[ Q = \frac{1}{N! \Lambda^{3N}} V^N \int d^{3N}q' \, e^{-\beta U(V^{1/3} q')} \]

3.2.5. Chemical Potential#

Also from ( Q ):

\[ \beta \mu = -\left( \frac{\partial \ln Q}{\partial N} \right)_{T, V} \]

Or:

\[ \mu = -k_B T \ln \left( \frac{Q(N+1)}{Q(N)} \right) \]

Example:

\[ \mu = -k_B T \ln \left( \frac{V}{(N+1) \Lambda^3} \right) \]

  • Excess chemical potential:

\[ \mu^{\text{ex}} = -k_B T \ln \left\langle e^{-\beta \Delta U} \right\rangle \]

Where ( \Delta U ) is the energy of a test particle inserted at a random position in the existing ensemble.

Note: Insertion can fail in dense systems. Methods like thermodynamic integration, fractional insertion, and Bennett’s acceptance ratio can help.

Note: Defining by deletion (rather than insertion) usually does not work. Pathological case: hard spheres.

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