🔢Analytic Combinatorics Unit 16 – Statistical Physics Applications

Key Concepts and Definitions

• Statistical physics applies probability theory and statistics to study the collective behavior of large systems of particles
• Microstates represent the detailed microscopic configurations of a system, while macrostates describe the system's overall macroscopic properties
• Ensemble is a collection of all possible microstates of a system, each assigned a probability based on the system's constraints
  • Microcanonical ensemble has fixed number of particles, volume, and energy
  • Canonical ensemble has fixed number of particles, volume, and temperature
  • Grand canonical ensemble has fixed chemical potential, volume, and temperature
• Partition function $Z$ is a sum over all possible microstates, weighted by their Boltzmann factors, and serves as a normalization constant for probability distributions
• Free energy $F$ is related to the partition function by $F = -k_B T \ln Z$ and determines the thermodynamic properties of a system
• Phase transitions occur when a system undergoes a dramatic change in its macroscopic properties (magnetization or density) as a control parameter (temperature or pressure) is varied
• Critical phenomena involve the behavior of a system near a phase transition, characterized by power-law divergences and universal scaling laws

Foundations of Statistical Physics

• Statistical mechanics provides a framework for relating the microscopic properties of a system to its macroscopic behavior
• Boltzmann distribution describes the probability of a system being in a particular microstate with energy $E_i$ at temperature $T$: $P(E_i) = \frac{1}{Z} e^{-E_i / k_B T}$
  • $k_B$ is the Boltzmann constant, and $Z$ is the partition function
• Entropy $S$ is a measure of the number of microstates available to a system and is given by $S = k_B \ln \Omega$, where $\Omega$ is the number of microstates
• Thermodynamic quantities (internal energy, pressure, and specific heat) can be derived from the partition function using statistical mechanics
• Equipartition theorem states that each quadratic degree of freedom in a system contributes $\frac{1}{2} k_B T$ to the average energy
• Fluctuations in macroscopic quantities (energy and magnetization) become relatively smaller as the system size increases, leading to the emergence of well-defined thermodynamic properties

Combinatorial Structures in Statistical Physics

• Combinatorial analysis is essential for enumerating the number of microstates in a system, which is crucial for calculating partition functions and thermodynamic properties
• Binomial coefficients $\binom{n}{k}$ count the number of ways to choose $k$ objects from a set of $n$ objects, relevant for systems with distinguishable particles
• Multinomial coefficients $\binom{n}{k_1, k_2, \ldots, k_m}$ generalize binomial coefficients to count the number of ways to partition $n$ objects into $m$ groups of sizes $k_1, k_2, \ldots, k_m$
• Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of $n$ objects into $k$ non-empty subsets, relevant for systems with indistinguishable particles
• Generating functions are powerful tools for analyzing combinatorial structures and can be used to derive asymptotic properties of partition functions
  • Ordinary generating functions $A(z) = \sum_{n \geq 0} a_n z^n$ encode the sequence $\{a_n\}$ in a formal power series
  • Exponential generating functions $B(z) = \sum_{n \geq 0} b_n \frac{z^n}{n!}$ are useful for problems involving labeled objects
• Polya's enumeration theorem provides a systematic way to count the number of distinct configurations under group actions, relevant for systems with symmetries

Probability Distributions and Ensembles

• Probability distributions assign probabilities to each microstate of a system based on the constraints imposed by the ensemble
• Microcanonical ensemble describes an isolated system with fixed number of particles $N$, volume $V$, and energy $E$, leading to a uniform probability distribution over all microstates with the given energy
• Canonical ensemble describes a system in thermal equilibrium with a heat bath at temperature $T$, leading to the Boltzmann distribution $P(E_i) = \frac{1}{Z} e^{-E_i / k_B T}$
  • Partition function for the canonical ensemble is $Z = \sum_i e^{-E_i / k_B T}$
• Grand canonical ensemble describes an open system that can exchange both energy and particles with a reservoir, characterized by fixed chemical potential $\mu$, volume $V$, and temperature $T$
  • Grand canonical partition function is $\Xi = \sum_{N=0}^\infty \sum_i e^{-(E_i - \mu N) / k_B T}$
• Gibbs distribution is a generalization of the Boltzmann distribution to systems with multiple conserved quantities, given by $P(\vec{x}) = \frac{1}{Z} e^{-\beta H(\vec{x})}$, where $H(\vec{x})$ is the Hamiltonian and $\beta = \frac{1}{k_B T}$
• Maximum entropy principle states that the probability distribution that best represents the current state of knowledge about a system is the one with the largest entropy, subject to the constraints imposed by the available information

Partition Functions and Free Energy

• Partition functions are central objects in statistical mechanics that encode the statistical properties of a system and allow the calculation of thermodynamic quantities
• For a system with discrete energy levels, the partition function is a sum over all microstates: $Z = \sum_i e^{-\beta E_i}$, where $\beta = \frac{1}{k_B T}$
• For a system with continuous degrees of freedom, the partition function involves an integral: $Z = \int e^{-\beta H(\vec{x})} d\vec{x}$, where $H(\vec{x})$ is the Hamiltonian
• Free energy is related to the partition function by $F = -k_B T \ln Z$ and determines the thermodynamic properties of the system
  • Helmholtz free energy $F = U - TS$ is used for systems with fixed temperature and volume
  • Gibbs free energy $G = U + PV - TS$ is used for systems with fixed temperature and pressure
• Thermodynamic quantities can be derived from the partition function using the following relations:
  • Internal energy: $U = -\frac{\partial \ln Z}{\partial \beta}$
  • Entropy: $S = k_B \ln Z + \beta U$
  • Pressure: $P = k_B T \frac{\partial \ln Z}{\partial V}$
• Partition functions for non-interacting systems can be factorized into a product of single-particle partition functions, greatly simplifying calculations
• Virial expansion expresses the equation of state of a gas as a power series in the density, with coefficients related to the partition function

Phase Transitions and Critical Phenomena

• Phase transitions occur when a system undergoes a qualitative change in its macroscopic properties as a control parameter (temperature or pressure) is varied
• First-order phase transitions are characterized by a discontinuity in the first derivative of the free energy (latent heat) and coexistence of phases (liquid-gas or solid-liquid transitions)
• Second-order (continuous) phase transitions are characterized by a continuous change in the order parameter (magnetization or density) and divergence of the correlation length (ferromagnetic or superconducting transitions)
• Critical point is the endpoint of a line of first-order phase transitions, where the distinction between phases vanishes and the system exhibits scale invariance and universality
• Order parameter is a quantity that distinguishes between the phases of a system and goes to zero at the critical point (magnetization for ferromagnets or density difference for liquid-gas transitions)
• Correlation length $\xi$ measures the typical size of fluctuations in the order parameter and diverges at the critical point as $\xi \sim |T - T_c|^{-\nu}$, where $\nu$ is a critical exponent
• Universality implies that the critical behavior of a system depends only on a few fundamental properties (symmetry and dimensionality) and not on the microscopic details
• Renormalization group theory provides a framework for understanding the scale invariance and universality of critical phenomena by studying how the system's properties change under coarse-graining transformations

Applications to Complex Systems

• Statistical physics has found wide-ranging applications in the study of complex systems, from condensed matter physics to biology, economics, and social sciences
• Ising model is a simple lattice model of ferromagnetism that exhibits a phase transition and has been used to study various phenomena, including opinion dynamics and neural networks
  • Spins can take values of $\pm 1$, and the energy of a configuration is given by $H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$, where $J$ is the interaction strength and $h$ is an external magnetic field
• Potts model generalizes the Ising model to spins with more than two states and has been used to study cell adhesion and tumor growth
• Percolation theory describes the formation of connected clusters in a random graph and has applications in material science (conductivity of composites) and epidemiology (spread of diseases)
• Random matrix theory studies the statistical properties of eigenvalues and eigenvectors of random matrices and has found applications in quantum chaos, nuclear physics, and financial markets
• Stochastic processes, such as random walks and diffusion, are ubiquitous in nature and can be analyzed using the tools of statistical physics
  • Fokker-Planck equation describes the time evolution of the probability density function for a stochastic process
  • Langevin equation is a stochastic differential equation that models the motion of a particle subject to random forces
• Complex networks, such as the Internet, social networks, and biological networks, exhibit non-trivial topological features (scale-free degree distributions and small-world properties) that can be studied using statistical physics methods

Problem-Solving Techniques and Examples

• Calculating partition functions:
  • For a system of $N$ non-interacting spins in a magnetic field $h$, the partition function is $Z = \left(2 \cosh \frac{h}{k_B T}\right)^N$
  • For a harmonic oscillator with frequency $\omega$, the partition function is $Z = \frac{1}{2 \sinh \frac{\hbar \omega}{2 k_B T}}$
• Deriving thermodynamic quantities:
  • For an ideal gas, the Helmholtz free energy is $F = -Nk_B T \ln \left(\frac{V}{N \lambda^3}\right)$, where $\lambda = \frac{h}{\sqrt{2 \pi m k_B T}}$ is the thermal wavelength
  • The pressure of an ideal gas is $P = \frac{Nk_B T}{V}$, and the entropy is $S = Nk_B \left(\ln \frac{V}{N \lambda^3} + \frac{5}{2}\right)$
• Analyzing phase transitions:
  • In the mean-field approximation of the Ising model, the magnetization satisfies the self-consistency equation $m = \tanh \frac{Jzm + h}{k_B T}$, where $z$ is the coordination number
  • The critical temperature is given by $k_B T_c = Jz$, and the critical exponents are $\beta = \frac{1}{2}$, $\gamma = 1$, and $\delta = 3$
• Solving stochastic processes:
  • For a random walk on a one-dimensional lattice with hopping rate $\gamma$, the diffusion coefficient is $D = \gamma a^2$, where $a$ is the lattice spacing
  • The mean square displacement grows linearly with time: $\langle x^2 \rangle = 2Dt$
• Applying combinatorial techniques:
  • The number of ways to distribute $N$ indistinguishable particles among $M$ energy levels, with $n_i$ particles in the $i$-th level, is given by the multinomial coefficient $\binom{N}{n_1, n_2, \ldots, n_M}$
  • The number of distinct configurations of $N$ particles on a lattice with $M$ sites, up to permutations, is given by the binomial coefficient $\binom{M+N-1}{N}$