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Statistical Physics of Complex Systems by Eric Bertin in 2016

!Statistical physics Bertin cover

books, statistical physics, Non-equilibrium statistical physics, stochastic processes

Author: Eric Bertin

1. Equilibrium statistical physics

2. Non-stationary dynamics and stochastic formalism

2.1 Markovian stochastic processes and master equation

  • 2.1.1 Definition of Markovian Stochastic Processes
  • 2.1.2 Master equation and detailed balance
  • 2.1.3 A simple example: the one-dimensional random walk

2.2 Langevin equation

  • 2.2.1 Phenomenological approach
  • 2.2.2 Basic properties of the linear Langevin equation
  • 2.2.3 More general forms of the Langevin equation
  • 2.2.4 Relation to random walks

2.3 Fokker-Planck equation

  • 2.3.1 Continuous limit of a discrete master equation
  • 2.3.2 Kramers-Moyal expansion
  • 2.3.3 More general forms of the Fokker-Planck equation

2.4 Anomalous diffusion: scaling arguments

  • Importance of the largest events
  • Superdiffusive random walks
  • Subdiffusive random walks

2.5 Fast and slow relaxation to equilibrium

  • 2.5.1 Relaxation to canonical equilibrium
  • 2.5.2 Dynamical increase of the entropy
  • 2.5.3 Slow relaxation and physical aging

3. Statistical physics of interacting macroscopic units

3.1 Dynamics of residential moves

  • 3.1.1 A simplified version of the Schelling model
  • 3.1.2 Condition for phase separation
  • 3.1.3 The true Schelling model: two types of agents

3.2 Driven particles on a lattice: zero-range process

  • 3.2.1 Definition and exact steady-state solution
  • 3.2.2 Maximal density and condensation phenomenon
  • 3.2.3 Dissipative zero-range process

3.3 Collective motion of active particles

  • 3.3.1 Derivation of continuous equations
  • 3.3.2 Phase diagram and instabilities
  • 3.3.3 Varying the symmetries of particles

4. Beyond assemblies of stable units

4.1 Non-conserved particles: reaction-diffusion processes

  • 4.1.1 Mean-field approach of absorbing phase transitions

    • Determine an evolution equation for the density field \[\rho(\mathbf{r},t)\]

      • Parameters

        • for each particle already present in the system, a new particle is created with probability \[\kappa\] per unit time

      • Reactions

        • \[A \longrightarrow 2A\] at rate \[\kappa\]
        • \[A \longrightarrow 0\] at rate \[\nu\]
        • \[2A \longrightarrow A\] at rate \[\lambda\]

      • Terms

        • Linear

          • the rate of change of the density resulting from the reaction \[A \longrightarrow 2A\] is \[\kappa \rho\]
          • the rate of change of the density resulting from the reaction \[A \longrightarrow 0\] is \[-\nu \rho\]
          • the rate of change of the density resulting from diffusion of particles \[A\] in space with a diffusion coefficient \[D\] is \[D \Delta \rho\] where \[\Delta\] is the Laplacian operator \[\Delta = \nabla^2\]

        • Nonlinear

          • an approximation to the rate of change of the density resulting from the reaction \[2A \longrightarrow A\] is \[-\lambda \rho^2\]

      • Dynamics

        • \[\frac{\partial \rho}{\partial t} = (\kappa * \nu)\rho * \lambda \rho^2 + D \Delta \rho\]
  • 4.1.2 Fluctuations in a fully connected model

4.2 Evolutionary dynamics

  • 4.2.1 Statistical physics modeling of evolution in biology
  • 4.2.2 Selection dynamics without mutation
  • 4.2.3 Quasistatic evolution under mutation

4.3 Dynamics of networks

  • 4.3.1 Random networks
  • 4.3.2 Small-world networks
  • 4.3.3 Preferential attachment

5. Statistical description of deterministic systems

5.1 Basic notions on deterministic systems

  • 5.1.1 Fixed points and simple attractors
  • 5.1.2 Bifurcations
  • 5.1.3 Chaotic dynamics

5.2 Deterministic versus stochastic dynamics key subsection

  • 5.2.1 Qualitative differences and similarities
  • 5.2.2 Stochastic coarse-grained description of a chaotic map
  • 5.2.3 Statistical description of chaotic systems

5.3 Globally coupled dynamical systems

  • 5.3.1 Coupling low-dimensional dynamical systems
  • 5.3.2 Description in terms of global order parameters
  • 5.3.3 Stability of the fixed point of the global system

5.4 Synchronization transition

  • 5.4.1 The Kuramoto model of coupled oscillators
  • 5.4.2 Synchronized steady state

6. A probabilistic viewpoint on fluctuations and rare events

6.1 Global fluctuations as a random sum problem

  • 6.1.1 Law of large numbers and central limit theorem
  • 6.1.2 Generalization to variable with infinite variances
  • 6.1.3 Case of non-identically distributed variables
  • 6.1.4 Case of correlated variables
  • 6.1.5 Coarse-graining procedures and law of large numbers

6.2 Rare and extreme events

  • 6.2.1 Different types of rare events
  • 6.2.2 Extreme value statistics
  • 6.2.3 Statistics of records

6.3 Large deviation functions key subsection

  • 6.3.1 A simple example: The Ising Model in a magnetic field
  • 6.3.2 Explicit computations of large deviation functions
  • 6.3.3 A natural framework to formulate statistical physics

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