Transport Phenomena and Phase Transition

When a system is driven out of thermodynamic equilibrium by temperature gradients, concentration gradients, or external electromagnetic fields, transport phenomena occur. The fundamental microscopic equation governing non-equilibrium statistical mechanics is the **Boltzmann Transport Equation** (Ludwig Boltzmann, 1872).

1. Phase-Space Distribution Function ($f(\mathbf{r}, \mathbf{p}, t)$)

Let $f(\mathbf{r}, \mathbf{p}, t) \, d^3\mathbf{r} \, d^3\mathbf{p}$ be the number of particles in physical volume element $d^3\mathbf{r}$ with momenta in $d^3\mathbf{p}$ at time $t$. In the absence of interparticle collisions, particles flow smoothly through phase space according to Hamilton's equations $\mathbf{v} = \mathbf{p}/m$ and $\dot{\mathbf{p}} = \mathbf{F}$: $$f(\mathbf{r} + \mathbf{v}\delta t, \, \mathbf{p} + \mathbf{F}\delta t, \, t + \delta t) = f(\mathbf{r}, \mathbf{p}, t)$$

2. The Collision Term

Interparticle collisions abruptly scatter particles into and out of the phase-space volume element: $$f(\mathbf{r} + \mathbf{v}\delta t, \, \mathbf{p} + \mathbf{F}\delta t, \, t + \delta t) - f(\mathbf{r}, \mathbf{p}, t) = \left( \frac{\partial f}{\partial t} \right)_{\text{coll}} \delta t$$ Expanding the left side in a first-order Taylor series yields the **Boltzmann Transport Equation**: $$\frac{\partial f}{\partial t} + \mathbf{v} \cdot \boldsymbol{\nabla}_{\mathbf{r}} f + \frac{\mathbf{F}}{m} \cdot \boldsymbol{\nabla}_{\mathbf{v}} f = \left( \frac{\partial f}{\partial t} \right)_{\text{coll}}$$

3. The Relaxation-Time (BGK) Approximation

Because the exact non-linear collision integral involves 5-dimensional integration over collision cross sections, Bhatnagar, Gross, and Krook (1954) introduced the widely used **Relaxation-Time Approximation**: $$\left( \frac{\partial f}{\partial t} \right)_{\text{coll}} \approx -\frac{f(\mathbf{r}, \mathbf{v}, t) - f_0(\mathbf{v})}{\tau}$$ where $f_0(\mathbf{v})$ is the local equilibrium Maxwell-Boltzmann distribution, and $\tau$ is the characteristic collision relaxation time. Any local perturbation decays exponentially back to equilibrium as $e^{-t/\tau}$.

How can time-reversible microscopic Newtonian mechanics give rise to irreversible macroscopic thermodynamics (Second Law)? Ludwig Boltzmann solved this with his famous **H-Theorem** (1872).

1. Definition of the $H$-Function

Boltzmann defined the functional $H(t)$ for a gas with velocity distribution $f(\mathbf{v}, t)$: $$H(t) \equiv \int f(\mathbf{v}, t) \ln f(\mathbf{v}, t) \, d^3\mathbf{v}$$ Because statistical entropy is $S = -k_B \int f \ln f \, d^3\mathbf{v}$, Boltzmann's $H$-function is directly proportional to negative entropy: $$S(t) = -k_B H(t) + \text{constant}$$

2. Derivation of $dH/dt \le 0$

Using the exact binary collision integral and the principle of detailed balance (microscopic time-reversibility of collisions $\mathbf{v}_1, \mathbf{v}_2 \leftrightarrow \mathbf{v}_1', \mathbf{v}_2'$): $$\frac{dH}{dt} = -\frac{1}{4} \int \dots \int (f_1' f_2' - f_1 f_2) \ln\left( \frac{f_1' f_2'}{f_1 f_2} \right) \sigma \, |\mathbf{v}_1 - \mathbf{v}_2| \, d\Omega \, d^3\mathbf{v}_1 \, d^3\mathbf{v}_2$$ Because for any positive numbers $x, y$, the inequality $(x - y)\ln(x/y) \ge 0$ holds strictly (being zero if and only if $x = y$): $$\frac{dH}{dt} \le 0 \iff \frac{dS}{dt} \ge 0$$

Boltzmann's H-Theorem: The functional $H(t)$ decreases monotonically over time until the gas attains the Maxwell-Boltzmann distribution, where $dH/dt = 0$. Consequently, macroscopic entropy can never decrease!

3. Resolution of Historical Paradoxes

• Loschmidt's Time-Reversal Paradox (Umkehreinwand): If Newton's laws are time-reversible, reversing all velocities should make entropy decrease. Resolution: The H-theorem assumes the Molecular Chaos Hypothesis (Stoßzahlansatz)—that velocities of colliding particles are uncorrelated before collision. Reversing velocities introduces exquisite fine-tuned microscopic correlations that violate this hypothesis.
• Zermelo's Recurrence Paradox (Wiederkehreinwand): Poincaré's theorem states any bounded mechanical system must return arbitrarily close to its initial microstate. Resolution: For $10^{23}$ particles, the Poincaré recurrence time is $\sim 10^{10^{23}}$ years—vastly longer than the age of the universe!

The **mean free path** $\lambda$ is the average distance traversed by a molecule between successive collisions.

1. Collision Cross Section

Model gas molecules as hard spheres of diameter $d$. A collision occurs whenever the centers of two molecules approach within distance $d$. The collision cross section is the area of a disk of radius $d$: $$\sigma = \pi d^2$$

2. Derivation of the Mean Free Path

If a molecule moves with relative speed $\bar{v}_{\text{rel}}$, in time $\Delta t$ its collision cross section sweeps out a cylinder of volume $\Delta V = \sigma \bar{v}_{\text{rel}} \Delta t$. If the number density of scatterers is $n = N/V$, the number of collisions is: $$\Delta N_{\text{coll}} = n \Delta V = n \pi d^2 \bar{v}_{\text{rel}} \Delta t$$ For a thermal gas, the average relative speed between two randomly oriented molecules is $\bar{v}_{\text{rel}} = \sqrt{2} \bar{v}$. The collision frequency is: $$\nu_{\text{coll}} = \frac{\Delta N_{\text{coll}}}{\Delta t} = \sqrt{2} \pi n d^2 \bar{v}$$ The mean free path is the average speed divided by collision frequency: $$\lambda = \frac{\bar{v}}{\nu_{\text{coll}}} = \frac{1}{\sqrt{2} \pi n d^2}$$ Using the ideal gas law $P = n k_B T$: $$\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$$ At room temperature and atmospheric pressure for air ($d \approx 3.7\text{ Å}$): $$\lambda \approx 68\text{ nm}$$ Molecules travel hundreds of times their own diameter between collisions!

The kinetic theory of transport explains how molecules transport momentum (viscosity), energy (thermal conductivity), and mass (diffusion).

1. Dynamic Viscosity ($\eta$): Momentum Transport

Consider a gas with a shear flow velocity gradient $\frac{du_x}{dy}$. Molecules crossing plane $y$ from a mean free path distance $\pm \lambda$ carry momentum $m u_x(y \pm \lambda)$. The shear stress is: $$\tau_{xy} = -\frac{1}{3} n m \bar{v} \lambda \frac{du_x}{dy} = -\eta \frac{du_x}{dy}$$ $$\eta = \frac{1}{3} \rho \bar{v} \lambda = \frac{1}{3} (n m) \bar{v} \left( \frac{1}{\sqrt{2} \pi n d^2} \right) = \frac{m \bar{v}}{3 \sqrt{2} \pi d^2} = \frac{\sqrt{m k_B T}}{3 \pi^{3/2} d^2}$$

Maxwell's Paradox: Notice that density $n$ cancels out completely! The viscosity of a dilute gas is completely independent of pressure and density! $$\left(\frac{\partial \eta}{\partial P}\right)_T = 0$$ Furthermore, gas viscosity increases with temperature ($\eta \propto \sqrt{T}$), unlike liquids whose viscosity drops with temperature.

2. Thermal Conductivity ($\kappa$): Heat Energy Transport

For a temperature gradient $\frac{dT}{dz}$, molecules transport thermal energy $c_v m T$: $$J_q = -\kappa \frac{dT}{dz}, \quad \kappa = \frac{1}{3} n c_v m \bar{v} \lambda = \frac{1}{3} C_V \rho \bar{v} \lambda = \frac{c_v \sqrt{m k_B T}}{3 \pi^{3/2} d^2}$$ Thermal conductivity is also independent of gas pressure.

3. Self-Diffusion Coefficient ($D$): Mass Transport

For a concentration gradient $\frac{dn}{dz}$, the particle flux is given by Fick's Law: $$J_N = -D \frac{dn}{dz}, \quad D = \frac{1}{3} \bar{v} \lambda = \frac{2}{3 \pi^{3/2} d^2} \frac{\sqrt{k_B T / m}}{n} \propto \frac{T^{3/2}}{P}$$ Unlike viscosity, diffusion is inversely proportional to pressure.

Electrical conduction in metals is the transport of electric charge under an applied electric field $\mathbf{E}$.

1. The Drude-Sommerfeld Conductivity Formula

In an electric field $\mathbf{E}$, conduction electrons experience acceleration $\mathbf{a} = -e\mathbf{E}/m^*$. Collisions with lattice phonons and impurities randomize momentum with relaxation time $\tau$. The steady-state drift velocity is: $$\mathbf{v}_d = -\frac{e \tau}{m^*} \mathbf{E}$$ The electric current density is given by Ohm's Law $\mathbf{J} = -n e \mathbf{v}_d = \sigma \mathbf{E}$, yielding: $$\sigma = \frac{n e^2 \tau}{m^*}$$

2. The Wiedemann-Franz Law and Lorenz Number

In metals, conduction electrons transport both electric charge and thermal heat. The electronic thermal conductivity is $\kappa_{\text{el}} = \frac{1}{3} C_V^{\text{el}} v_F^2 \tau$. Using the Sommerfeld linear electronic heat capacity $C_V^{\text{el}} = \frac{\pi^2}{2} n k_B (T / T_F)$: $$\kappa_{\text{el}} = \frac{1}{3} \left( \frac{\pi^2}{2} \frac{n k_B^2 T}{\epsilon_F} \right) \left( \frac{2\epsilon_F}{m^*} \right) \tau = \frac{\pi^2 n k_B^2 T \tau}{3 m^*}$$ Dividing thermal conductivity $\kappa$ by electrical conductivity $\sigma$: $$\frac{\kappa}{\sigma T} = \frac{\pi^2}{3} \left( \frac{k_B}{e} \right)^2 \equiv L$$ This is the celebrated **Wiedemann-Franz Law** (1853). The ratio is a universal physical constant known as the **Lorenz Number**: $$L = \frac{\pi^2}{3} \left( \frac{k_B}{e} \right)^2 = 2.443 \times 10^{-8}\text{ W}\cdot\Omega\text{/K}^2$$ This universal constant holds remarkably across copper, silver, gold, and aluminum at room temperature!

Brownian motion—the irregular, ceaseless jigging of microscopic particles suspended in a fluid—provided the definitive experimental proof of the physical reality of atoms.

1. The Langevin Equation (Paul Langevin, 1908)

A microscopic particle of mass $m$ and radius $a$ suspended in a fluid experiences two forces:

1. A macroscopic viscous drag force: $-\gamma v = -6\pi \eta a v$ (Stokes' Law).
2. A fluctuating, stochastic force $\xi(t)$ due to billions of thermal molecular collisions.

The stochastic equation of motion is: $$m \frac{dv}{dt} = -\gamma v + \xi(t)$$ The random force has zero mean $\langle \xi(t) \rangle = 0$ and is $\delta$-correlated in time (white noise): $$\langle \xi(t) \xi(t') \rangle = 2 \gamma k_B T \, \delta(t - t')$$ This is the **Fluctuation-Dissipation Theorem**: the microscopic random kicks $\xi(t)$ and the macroscopic viscous drag $\gamma$ have the exact same physical origin (collisions with fluid molecules).

2. Derivation of Mean Squared Displacement (Einstein, 1905)

Multiplying the Langevin equation by $x$ and taking ensemble averages: $$m \left\langle x \frac{dv}{dt} \right\rangle = -\gamma \langle x v \rangle + \langle x \xi \rangle$$ Using $\frac{d}{dt}(x^2) = 2xv$ and $\frac{d}{dt}(xv) = v^2 + x\frac{dv}{dt}$: $$\frac{m}{2} \frac{d^2}{dt^2} \langle x^2 \rangle - m \langle v^2 \rangle = -\frac{\gamma}{2} \frac{d}{dt} \langle x^2 \rangle$$ By the equipartition theorem, $m \langle v^2 \rangle = k_B T$. In the overdamped regime ($t \gg m/\gamma$): $$\frac{\gamma}{2} \frac{d}{dt} \langle x^2 \rangle = k_B T \implies \frac{d}{dt} \langle x^2 \rangle = \frac{2 k_B T}{\gamma}$$ Integrating over time: $$\langle x^2(t) \rangle = 2 D t$$ where $D$ is the **Einstein Diffusion Coefficient**: $$D = \frac{k_B T}{\gamma} = \frac{k_B T}{6\pi \eta a}$$ By tracking the displacement $\langle x^2 \rangle$ of gamboge particles under a microscope, Jean Perrin measured Boltzmann's constant $k_B$ and calculated Avogadro's number $N_A$, winning the 1926 Nobel Prize for proving the reality of atoms!

Phase transitions are transformations between distinct macroscopic states of matter (phases) as external thermodynamic parameters ($T, P, B$) vary.

1. The Ehrenfest Classification of Phase Transitions

Paul Ehrenfest (1933) classified phase transitions by the lowest order derivative of Gibbs free energy $G(T, P)$ that exhibits a mathematical discontinuity:

• First-Order Phase Transitions: Discontinuity in the first derivatives of $G$: $$\Delta S = -\Delta \left(\frac{\partial G}{\partial T}\right)_P \ne 0, \quad \Delta V = \Delta \left(\frac{\partial G}{\partial P}\right)_T \ne 0$$ Characterized by a non-zero **Latent Heat** $L = T \Delta S$ and a discontinuous change in volume/density $\Delta V$. Examples: Ice melting, water boiling, sublimation.
• Second-Order (Continuous) Phase Transitions: First derivatives are continuous (no latent heat: $\Delta S = 0, \Delta V = 0$), but second derivatives diverge or jump discontinuously: $$C_P = -T \left(\frac{\partial^2 G}{\partial T^2}\right)_P, \quad \kappa_T = -\frac{1}{V} \left(\frac{\partial^2 G}{\partial P^2}\right)_T, \quad \alpha = \frac{1}{V} \frac{\partial^2 G}{\partial T \partial P}$$ Examples: Ferromagnetic Curie transition, superconductor-normal transition in zero field, superfluid He-II $\lambda$-transition.

2. The Clausius-Clapeyron Equation

Along a first-order phase coexistence boundary curve $P(T)$, the two phases ($1$ and $2$) have identical chemical potential: $$dG_1 = dG_2 \implies -S_1 dT + V_1 dP = -S_2 dT + V_2 dP$$ Rearranging gives the **Clausius-Clapeyron Equation**: $$\frac{dP}{dT} = \frac{S_2 - S_1}{V_2 - V_1} = \frac{\Delta S}{\Delta V} = \frac{L}{T \Delta V}$$ Because ice expands when freezing ($V_{\text{ice}} > V_{\text{water}} \implies \Delta V < 0$), the melting curve of ice has a negative slope ($dP/dT < 0$), allowing ice skates to glide on a thin liquid layer!

The **Ising Model** (Ernst Ising and Wilhelm Lenz, 1925) is the preeminent model for studying collective phenomena, ferromagnetism, and continuous phase transitions.

1. The Microscopic Ising Hamiltonian

Consider a crystal lattice where each site $i$ has an Ising spin $s_i = \pm 1$ (spin up or spin down): $$H = -J \sum_{\langle i, j \rangle} s_i s_j - h \sum_i s_i$$ where $J > 0$ is the ferromagnetic exchange coupling between nearest neighbors $\langle i, j \rangle$, and $h = g \mu_B B$ is an external magnetic field.

2. Weiss Molecular Mean-Field Approximation

In Mean-Field Theory (MFT), we approximate the fluctuating neighbor spins $s_j$ by their average expectation value $\langle s \rangle = m$ (the **order parameter**): $$s_i s_j = (s_i - m + m)(s_j - m + m) \approx m s_i + m s_j - m^2$$ For a lattice with coordination number $z$ (number of nearest neighbors per site): $$H_{\text{MF}} = -\sum_i s_i (h + z J m) + \frac{1}{2} N z J m^2 = -\sum_i s_i h_{\text{eff}} + \frac{1}{2} N z J m^2$$ where $h_{\text{eff}} = h + z J m$ is the effective **Weiss molecular field**.

3. The Self-Consistency Equation

The single-spin partition function is: $$z_1 = e^{\beta h_{\text{eff}}} + e^{-\beta h_{\text{eff}}} = 2 \cosh(\beta h_{\text{eff}})$$ The average magnetization per site is: $$m = \langle s_i \rangle = \frac{e^{\beta h_{\text{eff}}} - e^{-\beta h_{\text{eff}}}}{z_1} = \tanh(\beta h_{\text{eff}})$$ Substituting $h_{\text{eff}} = h + z J m$: $$m = \tanh\left( \frac{z J m + h}{k_B T} \right)$$

4. Spontaneous Magnetization and Critical Curie Temperature

In zero external field ($h = 0$): $$m = \tanh\left( \frac{z J}{k_B T} m \right)$$ For high temperatures, the slope of $\tanh(x)$ at $x = 0$ is less than $1$: the only solution is $m = 0$ (paramagnetic phase). A non-trivial solution ($m \ne 0$) emerges when the slope at the origin exceeds $1$: $$\left.\frac{d}{dm}\tanh\left(\frac{z J m}{k_B T}\right)\right|_{m=0} = \frac{z J}{k_B T} > 1$$ This defines the **Curie Transition Temperature**: $$T_c = \frac{z J}{k_B}$$ For $T < T_c$, spontaneous symmetry breaking occurs: the spins spontaneously align into a macroscopic ferromagnet with $m \propto (T_c - T)^{1/2}$ near $T_c$ (mean-field critical exponent $\beta = 1/2$).