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Statistical Physics and Non-Equilibrium Transport

Manual: General · Subject: Applied Physics

Connect microstates to macroscopic observables, fluctuations, and transport phenomena in complex systems.

Ensembles and Thermodynamics

Partition function

The canonical partition function is Z=∑ie−βEiZ=\sum_i e^{-\beta E_i}, with β=1/(kBT)\beta=1/(k_B T). From ZZ, one obtains free energy, entropy, and response functions, creating a bridge between microscopic spectra and measurable thermodynamic properties.

Thermodynamic potentials

Helmholtz free energy FF
F=−kBTln⁡ZF=-k_B T\ln Z
Entropy SS
S=−(∂F/∂T)VS=-\left(\partial F/\partial T\right)_V
Internal energy UU
U=−∂ln⁡Z/∂βU=-\partial \ln Z/\partial \beta
Chemical potential μ\mu
Energy cost of particle exchange
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Fluctuation physics

In small systems, fluctuations can be comparable to mean values, so statistical distributions matter as much as averages.

Transport

Diffusion, electrical conduction, and heat flow can often be described by linear-response theory. For example, Fick's law gives J=−D∇n\mathbf{J}=-D\nabla n, while Fourier heat conduction gives q=−κ∇T\mathbf{q}=-\kappa\nabla T.

What is the canonical partition function used for?

Write the Boltzmann factor for a state of energy EiE_i.

From microscopic dynamics to diffusion

  1. 1

    Step 1: Define the microscopic random process or scattering mechanism.

  2. 2

    Step 2: Coarse-grain over many events to obtain a continuum density.

  3. 3

    Step 3: Derive a constitutive law such as J=−D∇n\mathbf{J}=-D\nabla n.

  4. 4

    Step 4: Combine with continuity to obtain the diffusion equation ∂tn=D∇2n\partial_t n=D\nabla^2 n.

Which law relates diffusive flux to concentration gradient?

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Advanced applications

Non-equilibrium statistical methods are essential in nanoscale heat transport, active matter, plasma kinetics, and stochastic thermodynamics.