Notes on Statistical Mechanics: Algorithm and Computations

book: Statistical mechanics
author: Werner Krauth

code: []

Table of Content

• Table of Content
  • Monte Carlo Methods
  • Hard Disks and Spheres
  • Density Matrices
  • The Bose Gas
  • Spin Systems
  • Entropic Forces
  • Dynamic MC Methods

Monte Carlo Methods

• physical concept: markov chains, detailed balance
• we set out to study a host of classical and quantum problems, all of value as models and with numerous applications and generalizations
• we also discuss the basic principles of statistical data analysis: how to extract results from well-behaved simulations
• Monte carlos is extremely general, and the basic recipes allow us - in principle - to solve any problem in statistical physics
• in practice, much effort has to be spent in designing algorithms specifically geared to the problem at hand
• concept of sampling
  • 2 fundamentally different sampling approaches: direct sampling and
  • markov-chain sampling: requires Metropolis algorithm
• Monte Carlo method is a powerful approach for the calculation of integrals
• narrative progresses from naive, brute-force sampling to the intelligent, guided exploration of configuration space that is the hallmark of Markov-chain Monte Carlo (MCMC)
• Fundamental concepts
  • direct sampling : generating statistically independent configurations from scracth, according to the target probability distribution
  • markov-chain sampling : generates sequence of correlated configurations, where each new state is a small, stochastic modification of the previous one
    • creates a “random walk” through the system’s configuration space
    • viability of this random walk depends on the rules that guarantee it explores the space correctly
      • cornerstone of these rules is the Metropolis algorithm:
        • correctness is ensured by the principle of detailed balance
        • in equilibrium: $\pi(a) P(a\rightarrow b) = \pi(b) P(b\rightarrow a)$
          • $\pi(x)$ is the equilibrium probability of configuration x
          • $P(x\rightarrow y)$ is the transition probability
        • this condition is sufficient (though not strictly necessary) to ensure that the Markov chain’s stationary distribution is the desired target distribution, such as the Boltzmann distribution $\pi(a) \propto e^{\beta E(a)}$
          • making this concrete: check the 3x3 pebble game
            • it illustrate detailed balance, irreducibility, and periodicity (3 conditions for a markov chain to converge to a unique stationary distribution)
  • Ergodicity: the principle that the Markov chain must be able to reach any physically relevant configuration from any other
  • Central limit theorem: the basis of estimating statistical errors and presents practical methods for handling the correlated data produced by Markov chains
    • bunching methods: to obtain reliable error estimates
• scientific mindset instilled: output of a computer simulation msut be treated with the same skepticism and subjected to the same rigorous analysis as data from a physical experiment

Hard Disks and Spheres

• core physical model: hard-sphere fluid
  • hard disks and spheres is used to bridge between the deterministic, time-evolving world of classical mechanics and the probabilistic, static framework of statistical mechanics
  • it has a crucial role in the historical development of both molecular dynamics and Monte Carlo methods
• physical concept: equiprobability, partition function
• 2 distinct descriptions of the same physical system
  • even-driven molecular dynamics (MD) : deterministic simulation of Newtonian mechanics
    • particles (position, velocities)
  • Boltzmann’s statistical mechanics : discards the notion of time and dynamics entirely
    • founded on a single, powerful axiom: principle of equiprobability
      • for an isolated system at constant energy, any two legal configurations are equally likely to be observed
      • using this statistical framework, core machinery of the displine called partition function $\mathcal{Z}$ is defined
        • the total volume of the allowed configuration space
      • its concrete, algorithmic meaning: the acceptance rate of a direct-sampling algorithm is directly proportional to the partition function
• Virial expansion: an analytical method for calculating the system’s equation of state at low densities, providing the theoretical benchmark against which simulation results can be tested
  • constant pressure, volume can fluctuate: $\pi(a) \propto e^{-\beta E(a)}$, where $E = PV$
• hard-disk system is rich enough to exhibit a phase transition
• ergodic hypothesis made concrete
• time-based simulation (molecular dynamics) vs. probabilistic simulation (monte carlo)
  • thermodynamic averaging over a long, deterministic trajectory vs. averaging over a large ensemble of static randomly generated configurations
  • both methods yield the same equation of state is a direct, numerical validation of the ergodic principle of this system
• the inherent chaotic dynamic of hard-sphere systems - where tiny perturbations in initial conditions lead to exponentially diverging trajectories further reinforces why the statistical approach is not only valid but essential for describing the system’s macroscopic behaviour
• algorithm as a scientific instrument
  • different algorithm can be interpreted as scientific probes that reveal fundamental properties of the physical system
    • algorithm as an instrument of investigation not just a tool for calculation
  • example: failure of the direct-sampling algorithm at high densities
  • “critical slowing down” observed in local Markov-chain Monte Carlo simulations near the freezing transition is a direct computational signature of a physical effect

Density Matrices

• core physical model: single quantum particle
• physical concept: feynman path integral
• thermal density matrix $\rho = e^{-\beta \hat{H}}$
  • quantum analog of the classical Boltzmann factor
    • $\hat{H}$: system’s Hamiltonian operator
    • $\beta = 1/k_B T$
• partition function : $Z = Tr(\rho)$, trace of the thermal density matrix
  • computational challenge: $\hat{T}$ and $\hat{V}$ do not commute, preventing separation of the exponential operator $e^{-\beta(\hat{T}+\hat{V})}$
    • solution: Trotter decomposition (Trotter-Suzuki formula)
      • allows one to approximate the low-temperature density matrix as a product of many high-temperature density matrices
      • this is the getaway to a computational treatment
    • When the Trotter formula is applied within the position-basis representation of the partition function, $Z = \int dx\bra{x} e^{(-\beta\hat{H}\ket{x})}$, a remarkable transformation occurs.
• convolution procedure
• establishes connection between classical and quantum systems -> basis of the Feynman path integral
• opens up the field of finite-temperature quantum statistics to computation
  • matrix squaring: we can express the density matrix at low temperature (\beta_1 + \beta_2) as an integral over density matrices at higher temperature (\beta_1, \beta_2) \begin{equation} \int dx \rho(x,x’,\beta_1) \rho(x’,x^{"},\beta_2) = \rho(x’,x^{"},\beta_1 + \beta_2) \end{equation}
• quantum partition function rewritten as a sum over classical-like trajectories in “imaginary time” is known as Feynman path integral - in this mapping: a single quantum particle is represented as classical “polymer” or “ring” - beads of the polymer corresponds to the particle’s position at discrete imaginary time slices - this mapping transforms the quantum simulation problem into a classical one, which can be solved with Monte Carlo Methods
• algorithm for sampling the configuration of these paths
  • efficient method: Levy construction
    • build the path in a hierarchical manner, correctly capturing the statistical properties of the underlying Brownian motion and dramatically improving sampling efficieny
• geometry of paths and related objects
  • will foster our understanding of quantum physics of the path integral

The Bose Gas

"what works once is a trick; what works twice is a method." ~ page 188

• core physical model: ideal bosons in a trap
• physical concept: bose-einstein condensation
• building upon the path-integral formalism for a single quantum particle, this chapter ventures into the real of quantum many-body systems
  • focus: ideal Bose gas, to explore the consequences of quantum indistinguishability
    • reveals an intuitive geometric interpretation of quantum statistics
    • culminates in a powerful approach to simulating Bose-Einstein condensation
  • challenge: simulating a system of identical bosons implementing the quantum statistical requirement that the total N-particle wave function must be symmetric under the exchange of any two particles.
    • the partition function must include a sum over all $N!$ permutations of the particle labels

Spin Systems

• core physical model: 2D ising model
• physical concepts: critical phenomena, phase transitions

Entropic Forces

• physical model: colloids, dimers
• physical concepts: depletion interaction, emergent order

Dynamic MC Methods

• physical model: sphere packing, RSD (?)
• physical concept: optimization, non-equilibrium dynamics