8.3 Transport phenomena

Transport phenomena describe how mass, energy, and momentum move through systems that are out of equilibrium. Within kinetic theory, these processes connect the microscopic behavior of individual gas molecules to macroscopic quantities you can measure, like thermal conductivity, viscosity, and diffusion rates.

Fundamentals of transport phenomena

Three types of transport show up repeatedly in kinetic theory: transport of mass (diffusion), transport of energy (heat conduction), and transport of momentum (viscosity). All three follow the same basic pattern: a macroscopic gradient drives a flux, and a transport coefficient sets the proportionality between them.

Diffusion vs convection

Diffusion is the net movement of particles from regions of high concentration to low concentration, driven entirely by random thermal motion. No bulk flow is needed. Convection, by contrast, carries substances or energy through bulk fluid motion, like wind carrying heat away from your skin.

• Diffusion dominates at microscopic scales and in still fluids
• Convection dominates at macroscopic scales and is generally much faster
• Many real systems involve both simultaneously

Microscopic vs macroscopic transport

At the microscopic level, transport is about individual molecules colliding, scattering, and carrying energy or momentum from one place to another. At the macroscopic level, you see smooth, averaged-out quantities like temperature profiles and flow velocities.

Statistical mechanics bridges these two scales. Brownian motion is a classic example: the erratic jiggling of a pollen grain in water is a macroscopic signature of microscopic molecular collisions.

Fluxes and gradients

A flux is the rate at which some quantity passes through a unit area per unit time. A gradient is the spatial rate of change of a property (concentration, temperature, velocity).

• In most transport phenomena, the relationship between flux and gradient is linear
• The flux direction opposes the gradient: heat flows from hot to cold, particles diffuse from high to low concentration
• This pattern is captured by the general form $J = -(\text{transport coefficient}) \times \nabla(\text{property})$

Diffusion processes

Diffusion occurs spontaneously whenever a concentration gradient exists, driven by the thermal motion of particles. It's the mechanism by which systems approach equilibrium and is central to non-equilibrium kinetic theory.

Fick's laws of diffusion

Fick's first law gives the diffusive flux for a steady-state concentration gradient:

$J = -D \frac{\partial c}{\partial x}$

The negative sign means particles flow from high to low concentration. $D$ is the diffusion coefficient, with units of $\text{m}^2/\text{s}$.

Fick's second law describes how the concentration profile evolves in time:

$\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$

This is a diffusion equation (mathematically identical in form to the heat equation). Both laws assume no bulk fluid motion or external forces.

Diffusion coefficient

The diffusion coefficient $D$ quantifies how quickly particles spread through a medium. It depends on:

• Temperature: higher $T$ means faster diffusion
• Particle size: larger particles diffuse more slowly
• Medium properties: viscosity, density, intermolecular interactions

The Einstein relation connects $D$ to particle mobility $\mu$:

$D = \mu k_B T$

This relation is powerful because it ties a macroscopic transport quantity to thermal energy at the molecular scale.

Random walk model

The simplest microscopic picture of diffusion is a random walk: a particle takes steps of random direction at regular time intervals. After many steps, the mean square displacement grows linearly with time:

$\langle x^2 \rangle = 2Dt$

(in one dimension; in three dimensions, $\langle r^2 \rangle = 6Dt$). This result connects the statistics of individual molecular trajectories directly to the macroscopic diffusion coefficient.

Heat conduction

Heat conduction is the transfer of thermal energy through matter without any bulk motion. It's the energy analog of particle diffusion: instead of particles spreading down a concentration gradient, thermal energy spreads down a temperature gradient.

Fourier's law

Fourier's law states that the heat flux is proportional to the temperature gradient:

$q = -k \frac{dT}{dx}$

Here $k$ is the thermal conductivity (units: $\text{W}/(\text{m} \cdot \text{K})$), and the negative sign ensures heat flows from hot to cold.

Thermal conductivity

Thermal conductivity measures how readily a material conducts heat. From kinetic theory of gases, it can be estimated as:

$k \sim \frac{1}{3} n \langle v \rangle \lambda c_v$

where $n$ is the number density, $\langle v \rangle$ is the mean molecular speed, $\lambda$ is the mean free path, and $c_v$ is the specific heat per molecule. Metals have high $k$ because free electrons carry energy efficiently alongside lattice vibrations.

Temperature gradients

Temperature gradients are the driving force for heat conduction. They can be:

• Steady-state: the temperature profile doesn't change with time (e.g., a rod with fixed temperatures at each end)
• Time-dependent: the profile evolves as the system equilibrates

Steep gradients produce large heat fluxes; as the system approaches thermal equilibrium, gradients flatten and heat flow slows.

Viscous flow

Viscosity is the transport of momentum between adjacent fluid layers moving at different velocities. When faster-moving molecules migrate into a slower layer (or vice versa), they carry momentum with them, producing an internal friction force.

Newton's law of viscosity

For a Newtonian fluid, the shear stress is proportional to the velocity gradient:

$\tau = \mu \frac{du}{dy}$

Here $\mu$ is the dynamic viscosity (units: $\text{Pa} \cdot \text{s}$), $u$ is the flow velocity, and $y$ is the direction perpendicular to flow. Water and air are approximately Newtonian.

Shear stress and strain rate

• Shear stress ($\tau$): the force per unit area acting parallel to the flow direction
• Strain rate ($du/dy$): the rate at which the fluid deforms

For Newtonian fluids, the relationship is linear. Non-Newtonian fluids (blood, polymer solutions, ketchup) show nonlinear relationships: their effective viscosity changes with strain rate.

Navier-Stokes equations

The Navier-Stokes equations govern the motion of viscous fluids:

$\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g}$

Each term has a physical meaning:

• $\rho \frac{D\mathbf{u}}{Dt}$: inertial forces (mass × acceleration of a fluid element)
• $-\nabla p$: pressure gradient forces
• $\mu \nabla^2 \mathbf{u}$: viscous forces
• $\rho \mathbf{g}$: body forces (gravity)

These equations can, in principle, be derived from the Boltzmann equation by taking appropriate moments.

Mass transport

Mass transport describes the movement of chemical species within a system. It's closely related to diffusion but often occurs alongside heat transfer and fluid flow in realistic situations.

Concentration gradients

Concentration gradients are spatial variations in the amount of a chemical species. They arise from:

• Chemical reactions consuming or producing species locally
• Phase changes (evaporation, dissolution)
• External inputs or boundary conditions

These gradients drive diffusive mass transport and directly influence reaction rates and equilibrium positions.

Fick's first law application

Fick's first law in the mass transport context:

$J = -D \frac{dc}{dx}$

This applies to steady-state diffusion where the concentration profile isn't changing with time. Practical applications include membrane transport (gas separation, dialysis) and drug delivery systems where a sustained concentration gradient drives controlled release. For multi-component mixtures, cross-diffusion effects can arise where the gradient of one species drives flux of another.

Diffusion in gases vs liquids

Coupled transport phenomena

Real systems rarely involve just one type of transport. When multiple gradients exist simultaneously, cross-effects appear: a temperature gradient can drive mass flow, an electric field can drive heat flow, and so on. The framework of irreversible thermodynamics handles these couplings systematically.

Thermoelectric effects

• Seebeck effect: a temperature difference across a conductor generates a voltage. This is how thermocouples work.
• Peltier effect: passing current through a junction of two materials causes heating or cooling. Used in solid-state coolers.
• Thomson effect: heat is absorbed or released when current flows through a conductor with a temperature gradient.

All three are manifestations of coupled charge and heat transport.

Electrokinetic phenomena

These arise from coupling between electric fields and fluid flow in systems with charged surfaces:

• Electrophoresis: charged particles migrate in an applied electric field (used in gel electrophoresis for DNA separation)
• Electroosmosis: an electric field drives bulk fluid flow through a charged capillary
• Streaming potential: forcing fluid through a charged channel generates a voltage

These effects are central to microfluidics and colloidal science.

Thermophoresis

Thermophoresis is the directed motion of particles along a temperature gradient (typically from hot to cold in gases). The effect depends on particle size, shape, and the surrounding fluid properties. Applications include thermal field-flow fractionation for particle separation and studies of prebiotic molecular concentration.

Statistical mechanics approach

Kinetic theory provides the microscopic foundation for all the transport laws above. Rather than postulating Fourier's law or Fick's law, you can derive them from the dynamics of molecular collisions.

Boltzmann equation

The Boltzmann equation governs the evolution of the single-particle distribution function $f(\mathbf{r}, \mathbf{v}, t)$ in phase space:

$\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \frac{\mathbf{F}}{m} \cdot \nabla_v f = C[f]$

• The left side describes free streaming and the effect of external forces
• The collision term $C[f]$ on the right accounts for molecular collisions that redistribute velocities
• At equilibrium, $C[f] = 0$ and $f$ reduces to the Maxwell-Boltzmann distribution

Chapman-Enskog theory

Chapman-Enskog theory is a systematic perturbation method for solving the Boltzmann equation when the system is close to local equilibrium.

1. Expand the distribution function as $f = f^{(0)} + \epsilon f^{(1)} + \epsilon^2 f^{(2)} + \cdots$, where $\epsilon$ is proportional to the Knudsen number (ratio of mean free path to system size)
2. The zeroth-order term $f^{(0)}$ is the local Maxwell-Boltzmann distribution
3. The first-order correction $f^{(1)}$ yields expressions for viscosity, thermal conductivity, and diffusion in terms of collision integrals
4. Higher-order terms give corrections for more rarefied conditions

This theory is most accurate for dilute gases where binary collisions dominate.

Mean free path concept

The mean free path $\lambda$ is the average distance a molecule travels between successive collisions:

$\lambda = \frac{1}{n \sigma \sqrt{2}}$

where $n$ is the number density and $\sigma$ is the collision cross-section. The mean free path sets the scale for all transport coefficients in a gas. When $\lambda$ is much smaller than the system size, the continuum (fluid) description is valid. When $\lambda$ is comparable to the system size (high Knudsen number), you need kinetic theory directly.

Transport coefficients

All three major transport coefficients for a dilute gas can be estimated from kinetic theory using the mean free path, mean speed $\langle v \rangle$, and relevant molecular properties.

Viscosity coefficient

$\mu \sim \frac{1}{3} n m \langle v \rangle \lambda$

where $m$ is the molecular mass. A key prediction of kinetic theory: for an ideal gas, viscosity is independent of pressure (because $n$ and $\lambda$ have opposite pressure dependences that cancel). Viscosity of gases increases with temperature (more energetic collisions transfer more momentum), while viscosity of liquids decreases with temperature.

Thermal conductivity coefficient

$k \sim \frac{1}{3} n c_v \langle v \rangle \lambda$

Like viscosity, $k$ for an ideal gas is approximately independent of pressure. In solids, thermal conductivity includes contributions from both lattice vibrations (phonons) and free electrons. The temperature dependence can be complex, especially in crystalline solids where phonon scattering mechanisms change with temperature.

Diffusion coefficient derivation

From kinetic theory, the self-diffusion coefficient is:

$D \sim \frac{1}{3} \langle v \rangle \lambda$

This can also be derived more rigorously from the Boltzmann equation via Chapman-Enskog theory, yielding results in terms of collision integrals that depend on the intermolecular potential. The diffusion coefficient scales as $D \propto T^{3/2}/P$ for hard-sphere molecules, reflecting the temperature dependence of both $\langle v \rangle$ and $\lambda$.

Irreversible thermodynamics

Classical thermodynamics handles equilibrium states. Irreversible thermodynamics extends the framework to systems with gradients and fluxes, under the assumption that each small region is in local equilibrium even though the system as a whole is not.

Onsager reciprocal relations

In the linear regime (small deviations from equilibrium), each flux $J_i$ depends linearly on all the thermodynamic forces $X_j$:

$J_i = \sum_j L_{ij} X_j$

The Onsager reciprocal relations state that the matrix of transport coefficients is symmetric:

$L_{ij} = L_{ji}$

This symmetry is not obvious from macroscopic physics. It follows from microscopic reversibility (time-reversal symmetry of the underlying molecular dynamics). These relations reduce the number of independent transport coefficients you need to measure for coupled phenomena.

Entropy production

In an irreversible process, entropy is produced at a rate:

$\dot{S} = \sum_i J_i X_i \geq 0$

This expression connects fluxes and forces to the second law of thermodynamics. The inequality ensures that spontaneous processes always increase total entropy. Entropy production provides a variational principle: in some cases, steady states correspond to minimum entropy production (Prigogine's theorem, valid in the linear regime).

Linear response theory

Linear response theory describes how a system responds to a small external perturbation. The central result is the fluctuation-dissipation theorem: the way a system dissipates energy when perturbed is directly related to its spontaneous thermal fluctuations at equilibrium.

• A system that fluctuates a lot at equilibrium will also respond strongly to perturbations
• This connects equilibrium statistical mechanics to non-equilibrium transport
• It provides a route to calculating transport coefficients from equilibrium simulations

Applications in statistical mechanics

Brownian motion

Brownian motion is the random, erratic movement of mesoscopic particles (like pollen grains or colloidal spheres) suspended in a fluid. It arises from the cumulative effect of countless molecular collisions.

• The Langevin equation models this by adding a random force term to Newton's second law: $m\ddot{x} = -\gamma \dot{x} + \eta(t)$, where $\gamma$ is the friction coefficient and $\eta(t)$ is a random (white noise) force
• On long time scales, the motion becomes diffusive: $\langle x^2 \rangle = 2Dt$
• Einstein showed that $D = k_B T / \gamma$, connecting the diffusion coefficient to thermal fluctuations and friction

Fluctuation-dissipation theorem

The fluctuation-dissipation theorem relates the imaginary part of the response function $\chi''(\omega)$ to the spectral density of fluctuations $S(\omega)$:

$\chi''(\omega) = \frac{\omega}{2k_BT} S(\omega)$

This means you can predict how a system will respond to an external drive just by measuring its equilibrium noise spectrum. The theorem applies near thermal equilibrium and is foundational to understanding noise in electrical circuits, mechanical oscillators, and many other systems.

Green-Kubo relations

Green-Kubo relations express transport coefficients as time integrals of equilibrium correlation functions:

$L = \frac{1}{k_BT} \int_0^\infty \langle J(t) J(0) \rangle \, dt$

Here $J(t)$ is the microscopic flux (of energy, momentum, or particles) and the angle brackets denote an equilibrium ensemble average. For example, viscosity can be computed from the autocorrelation of the stress tensor, and thermal conductivity from the autocorrelation of the heat current.

These relations are the basis for computing transport coefficients in molecular dynamics simulations, where you run an equilibrium simulation and extract transport properties from the time correlations of the appropriate fluxes.