TRB PG Assistant - Phase space and Liouville's theorem, microcanonical ensemble, statistical equations, thermodynamic functions of an ideal gas, equipartition of energy E-Book Study Material

```html

Phase Space and Liouville's Theorem

In statistical mechanics, we often deal with systems that have a very large number of particles. It's impossible to track the exact position and momentum of each individual particle. Instead, we use a probabilistic approach. To do this, we need a way to represent the state of the entire system at any given moment. This is where the concept of phase space comes in.

Phase Space

Phase space is a multi-dimensional abstract space where each dimension represents a degree of freedom of the system. For a single particle moving in one dimension, its state can be described by its position (q) and its momentum (p). So, the phase space for this single particle would be a 2-dimensional space with axes q and p.

For a system of N particles in three dimensions, each particle has 3 position coordinates (x, y, z) and 3 momentum coordinates (p_x, p_y, p_z). Therefore, a system of N particles has 3N position coordinates and 3N momentum coordinates. The total number of dimensions in the phase space for such a system is 6N. A single point in this 6N-dimensional phase space represents the instantaneous state of the entire system.

As the system evolves over time, the point representing its state traces out a trajectory in phase space.

Liouville's Theorem

Liouville's theorem is a fundamental result in classical statistical mechanics. It states that the density of states in phase space remains constant along the trajectory of a system. In simpler terms, if you consider a collection of systems (an ensemble) starting from different initial states but evolving under the same Hamiltonian, the volume occupied by these systems in phase space remains constant over time.

Mathematically, let ρ(q, p, t) be the probability density function in phase space. Liouville's theorem states that the total time derivative of ρ is zero:

$\frac{d\rho}{dt} = \frac{\partial\rho}{\partial t} + \sum_{i=1}^{f} \left(\frac{\partial\rho}{\partial q_i}\frac{dq_i}{dt} + \frac{\partial\rho}{\partial p_i}\frac{dp_i}{dt}\right) = 0$

where f is the number of degrees of freedom (f = 6N for N particles in 3D), q_i are the generalized coordinates, and p_i are the corresponding generalized momenta.

The terms $\frac{dq_i}{dt}$ and $\frac{dp_i}{dt}$ are related to the Hamiltonian of the system by Hamilton's equations of motion:

$\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}}$

$\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}}$

Substituting these into the total time derivative equation, we get:

$\frac{d\rho}{dt} = \frac{\partial\rho}{\partial t} + \sum_{i=1}^{f} \left(\frac{\partial\rho}{\partial q_i}\frac{\partial H}{\partial p_i} - \frac{\partial\rho}{\partial p_i}\frac{\partial H}{\partial q_i}\right) = 0$

The term in the parenthesis is the Poisson bracket {H, ρ}. So, the equation can be written as:

$\frac{\partial\rho}{\partial t} + \{H, \rho\} = 0$

This means that if the probability density is not explicitly dependent on time ( $\frac{\partial\rho}{\partial t} = 0$ ), then the density is conserved along the trajectories.

Key Takeaway: Liouville's theorem implies that the "fluid" of representative points in phase space behaves like an incompressible fluid. If a small region in phase space expands in some dimensions, it must contract in others to maintain a constant volume. This is crucial for establishing equilibrium statistical mechanics.

Microcanonical Ensemble

An ensemble is a collection of a large number of identical systems that are macroscopically indistinguishable but microscopically different. We use ensembles to calculate the average properties of a system.

The microcanonical ensemble is one of the fundamental ensembles in statistical mechanics. It represents an isolated system, meaning it does not exchange energy or particles with its surroundings.

Key characteristics of a system in the microcanonical ensemble:

Fixed Energy (E): The total energy of the system is constant.
Fixed Volume (V): The system is confined to a fixed volume.
Fixed Number of Particles (N): The number of particles in the system is constant.

According to the postulates of statistical mechanics, for an isolated system in equilibrium, all accessible microstates corresponding to a given total energy E are equally probable.

If $\Omega(E, V, N)$ is the number of microstates with energy between E and E + δE (where δE is a small energy range), then the probability of finding the system in any one of these microstates is:

$P_i = \frac{1}{\Omega(E, V, N)}$ for each accessible microstate i.

The total number of microstates Ω is related to the entropy (S) of the system by Boltzmann's famous formula:

$S = k_B \ln \Omega$

where k_B is Boltzmann's constant. This equation is a cornerstone of statistical mechanics, bridging the microscopic world of states and the macroscopic world of thermodynamics.

Memory Trick: Think of "Microcanonical" as "Microscopic" and "Canonical" meaning "rule" or "standard". It's the standard way to describe a truly isolated system at the microscopic level. Also, "Micro" often relates to smallness, and here it refers to the smallest possible energy fluctuations because the system is isolated.

Statistical Equations and Thermodynamic Functions of an Ideal Gas

An ideal gas is a theoretical gas composed of point particles that move randomly and do not interact with each other, except through perfectly elastic collisions. Despite its simplicity, the ideal gas model provides a good approximation for the behavior of many real gases at low pressures and high temperatures.

We use ensembles to derive the thermodynamic properties of an ideal gas. For an ideal gas, the Hamiltonian is simply the sum of the kinetic energies of the individual particles, as there are no potential energy terms due to interactions.

For N particles, each of mass m, moving in a volume V, the Hamiltonian is:

$H = \sum_{i=1}^{N} \frac{\vec{p}_i^2}{2m} = \sum_{i=1}^{N} \left(\frac{p_{ix}^2 + p_{iy}^2 + p_{iz}^2}{2m}\right)$

The Canonical Ensemble

While the microcanonical ensemble describes isolated systems, the canonical ensemble is more convenient for deriving thermodynamic properties when a system is in thermal contact with a heat reservoir at a constant temperature T. In this ensemble, the system can exchange energy with the reservoir, but its volume and number of particles are fixed.

The probability of a system being in a particular microstate with energy E_i is given by the Boltzmann distribution:

$P_i = \frac{e^{-E_i / k_B T}}{Z}$

where k_B is Boltzmann's constant, T is the absolute temperature, and Z is the partition function. The partition function is a normalization constant and is defined as the sum of the Boltzmann factors over all possible microstates:

$Z = \sum_i e^{-E_i / k_B T}$

Partition Function for an Ideal Gas

For a system of N non-interacting identical particles (an ideal gas), the total energy is the sum of individual particle energies. This simplifies the calculation of the partition function.

First, consider the partition function for a single particle (z₁):

$z_1 = \sum_{\text{states of one particle}} e^{-E_i / k_B T}$

We can treat the sum over states as an integral over phase space. For a single particle in volume V, the integral is:

$z_1 = \frac{1}{h^3} \int e^{-p^2 / 2mk_B T} d^3p \int d^3r$

Here, h is Planck's constant (introduced for quantum mechanical reasons, though we are primarily dealing with classical mechanics here, it provides the correct units and scale), $d^3p = dp_x dp_y dp_z$ and $d^3r = dx dy dz$ .

The spatial integral $\int d^3r$ over the volume V simply gives V.

The momentum integral is a standard Gaussian integral:

$\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}}$

For the momentum integral, a = 1 / (2mk_BT). So, the integral over $d^3p$ is:

$\int_{-\infty}^{\infty} e^{-(p_x^2 + p_y^2 + p_z^2) / 2mk_B T} dp_x dp_y dp_z = \left(\sqrt{2\pi mk_B T}\right)^3 = (2\pi mk_B T)^{3/2}$

Therefore, the single-particle partition function is:

$z_1 = \frac{V}{h^3} (2\pi mk_B T)^{3/2}$

For N identical, indistinguishable particles, the system partition function Z is related to z₁ by:

$Z = \frac{z_1^N}{N!}$

The N! in the denominator accounts for the indistinguishability of the particles (Gibbs paradox correction).

Substituting z₁:

$Z = \frac{1}{N!} \left(\frac{V}{h^3} (2\pi mk_B T)^{3/2}\right)^N = \frac{V^N}{N! h^{3N}} (2\pi mk_B T)^{3N/2}$

Thermodynamic Functions of an Ideal Gas

Once we have the partition function, we can derive all macroscopic thermodynamic properties.

Helmholtz Free Energy (F):
$F = -k_B T \ln Z$

Substituting Z:

$F = -k_B T \ln \left(\frac{V^N}{N! h^{3N}} (2\pi mk_B T)^{3N/2}\right)$

Using Stirling's approximation for ln(N!) ≈ N ln N - N:

$F = -N k_B T \left(1 + \ln\left(\frac{V}{N h^3}\right) + \frac{3}{2}\ln(2\pi mk_B T)\right)$
Internal Energy (U):
$U = -\left(\frac{\partial \ln Z}{\partial \beta}\right)_V$ where β = 1 / k_BT.

Alternatively, and more directly from F:

$U = F + TS$ and S = -(\partial F / \partial T)_V.

From F = -N k_B T \ln(\dots), we find the internal energy to be:

$U = \frac{3}{2} N k_B T$

This result aligns perfectly with the equipartition theorem.
Entropy (S):
$S = -\left(\frac{\partial F}{\partial T}\right)_{V,N}$

$S = N k_B \left(\frac{3}{2} + \ln\left(\frac{V}{N h^3}\right) + \frac{3}{2}\ln(2\pi mk_B T)\right)$

This is the Sackur-Tetrode equation for the entropy of a monatomic ideal gas.
Pressure (P):
$P = -\left(\frac{\partial F}{\partial V}\right)_{T,N}$

$P = \frac{N k_B T}{V}$

Rearranging gives the ideal gas law: PV = Nk_BT.

Ideal Gas Law Derivation Shortcut: Remember that the partition function Z for N non-interacting particles is Z = (z₁)^N / N!. The pressure P is directly related to the derivative of ln Z with respect to V. Since z₁ is proportional to V, ln Z is proportional to N ln V. Taking the derivative with respect to V gives P proportional to N/V, and the constant of proportionality comes from the temperature dependence, leading to PV = Nk_BT.

Equipartition of Energy

The equipartition theorem is a classical statistical mechanics result that states that, in thermal equilibrium, each degree of freedom that appears quadratically in the Hamiltonian contributes an average energy of k_BT / 2 per particle.

Degrees of Freedom

A degree of freedom refers to an independent parameter that specifies the configuration of a system. For a particle, these are typically its coordinates.

Translational degrees of freedom: These correspond to the motion of the center of mass of the particle or molecule. A free particle in 3D space has 3 translational degrees of freedom (along x, y, and z axes). The kinetic energy associated with these is (1/2)mv_x², (1/2)mv_y², and (1/2)mv_z². In terms of momentum, it's p_x² / 2m, p_y² / 2m, and p_z² / 2m.
Rotational degrees of freedom: These correspond to the rotation of a molecule about its center of mass. A linear molecule (like O₂ or CO) has 2 rotational degrees of freedom. A non-linear molecule (like H₂O) has 3 rotational degrees of freedom. The energy associated with rotation is also quadratic in angular velocity or angular momentum.
Vibrational degrees of freedom: If a molecule is treated as a system of atoms connected by springs, the vibrations of these atoms relative to each other contribute vibrational degrees of freedom. Each vibrational mode contributes both kinetic and potential energy terms that are quadratic in the vibrational coordinate and momentum. A single vibrational mode contributes k_BT on average (one k_BT / 2 from kinetic energy and one k_BT / 2 from potential energy).

Statement of the Theorem

The equipartition theorem states that if the Hamiltonian of a system can be written as a sum of terms, each depending on a single coordinate or momentum squared, then the average value of each such term in thermal equilibrium is k_BT / 2.

Consider a term in the Hamiltonian of the form A x² or B p², where x or p is a coordinate or momentum and A or B is a positive constant. The average value of this term is given by:

$\langle A x^2 \rangle = \frac{\int A x^2 e^{-A x^2 / k_B T} dx}{\int e^{-A x^2 / k_B T} dx}$

Using standard Gaussian integral results (∫ x² e^-ax² dx = (1/2a)√(π/a) and ∫ e^-ax² dx = √(π/a)), with a = A / k_BT:

$\langle A x^2 \rangle = A \frac{\frac{1}{2(A/k_B T)}\sqrt{\frac{\pi k_B T}{A}}}{\sqrt{\frac{\pi k_B T}{A}}} = A \frac{k_B T}{2A} = \frac{k_B T}{2}$

Similarly, for a momentum term B p², the average kinetic energy is B 2> = k_BT / 2.

Applications to Ideal Gases

Let's apply this to different types of ideal gases:

Monatomic Ideal Gas (e.g., He, Ne, Ar):
Molecules are single atoms. They have 3 translational degrees of freedom. At moderate temperatures, rotation and vibration are not significantly excited.

Average energy per molecule = 3 × (k_BT / 2) = (3/2)k_BT.

This matches the internal energy U = (3/2)Nk_BT derived from the partition function.

The molar heat capacity at constant volume (C_V) is the energy change per mole per degree Kelvin. For 1 mole, N = N_A (Avogadro's number), and N_Ak_B = R (the universal gas constant).

U = (3/2)Nk_BT

C_V = (∂U/∂T)_V / mole = (3/2)R.
Diatomic Ideal Gas (e.g., H₂, N₂, O₂):
Molecules have 3 translational degrees of freedom and 2 rotational degrees of freedom (rotation about axes perpendicular to the bond). Vibration is often not significant at room temperature.

At moderate temperatures (room temperature): 3 translational + 2 rotational = 5 degrees of freedom.

Average energy per molecule = 5 × (k_BT / 2) = (5/2)k_BT.

Molar heat capacity C_V = (5/2)R.

At high temperatures: Vibration becomes active. Each vibrational mode contributes k_BT (kinetic + potential). So, 3 translational + 2 rotational + 2 vibrational (for a diatomic molecule's internal motion) = 7 degrees of freedom.

Average energy per molecule = 7 × (k_BT / 2) = (7/2)k_BT.

Molar heat capacity C_V = (7/2)R.

Note: The vibrational modes are excited at higher temperatures due to quantum effects; the equipartition theorem strictly applies in the classical limit. However, it provides a good framework for understanding the trends.
Polyatomic (Non-linear) Ideal Gas (e.g., H₂O):
3 translational + 3 rotational degrees of freedom.

At moderate temperatures: 6 degrees of freedom.

Average energy per molecule = 6 × (k_BT / 2) = 3k_BT.

Molar heat capacity C_V = 3R.

Equipartition Summary:

Translational: 3 degrees of freedom (contributes 3k_BT/2 per molecule)
Rotational (Linear): 2 degrees of freedom (contributes 2k_BT/2 per molecule)
Rotational (Non-linear): 3 degrees of freedom (contributes 3k_BT/2 per molecule)
Vibrational: Each mode contributes k_BT per molecule (kinetic + potential)

The total energy per molecule is the sum of contributions from all active degrees of freedom.

Limitations of Equipartition Theorem

The equipartition theorem is a classical result and breaks down at low temperatures where quantum effects become important. For example, rotational and vibrational degrees of freedom are "frozen out" at low temperatures, meaning they do not contribute to the heat capacity because the energy quanta required to excite them are larger than the available thermal energy (k_BT).

The theorem assumes that all degrees of freedom contribute quadratically to the Hamiltonian. This is true for simple harmonic oscillators and free particles but may not hold for more complex systems.

```