:orphan:

Examples
========

Runnable demonstrations of the central results in each
``physicskit.statphys`` chapter: lattice phase transitions, molecular
dynamics, random walks, and self-organized criticality.

See also the narrative tutorial:

- :doc:`/tutorials/statphys_deep_dives`

Each script in this gallery is self-contained and can be run directly with
``python examples/statphys/<section>/<script>.py``. Every script also carries
an RST module docstring as its title/description and uses ``# %%`` markers to
split narrative text from code, which is exactly what Sphinx-Gallery renders
into the pages below -- the script *is* the source of truth for what you see,
not a copy of it.

Sections
--------

- **ising** -- the 2D Ising model: Onsager's exact phase transition,
  finite-size scaling to extract critical exponents, and critical slowing
  down (Metropolis versus the Wolff cluster algorithm).
- **xy_model** -- the Kosterlitz-Thouless transition via vortex-antivortex
  unbinding.
- **potts_model** -- first- versus second-order transitions in the q-state
  Potts model.
- **spin_glass** -- frustration and the Edwards-Anderson order parameter.
- **percolation** -- percolation as a purely geometric phase transition.
- **renormalization** -- Kadanoff block-spin renormalization group flow.
- **quantum_statistics** -- Bose-Einstein and Fermi-Dirac distributions and
  Bose-Einstein condensation.
- **molecular_dynamics** -- Boltzmann's H-theorem: relaxation to the
  Maxwell-Boltzmann distribution.
- **random_walk** -- diffusion, the Einstein relation, and the central limit
  theorem.
- **ehrenfest_urn** -- the Ehrenfest urn: reversibility, recurrence, and the
  arrow of time.
- **sandpile** -- self-organized criticality in the Bak-Tang-Wiesenfeld
  sandpile.
- **interactive** -- interactive Plotly dashboards.


.. raw:: html

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Ehrenfest urn
-------------

Reversible microscopic dynamics and the emergence of an arrow of time.


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    <div class="sphx-glr-thumbcontainer" tooltip="Two objections were raised against Boltzmann&#x27;s H-theorem soon after he proposed it: Loschmidt argued that time-reversible microscopic dynamics cannot produce genuinely irreversible macroscopic behavior, and Zermelo noted that any bounded mechanical system must (by Poincare recurrence) eventually return arbitrarily close to its initial state. The Ehrenfests&#x27; 1907 urn model resolves both at once, in the simplest possible setting.">

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  .. image:: /api/gallery/statphys/ehrenfest_urn/images/thumb/sphx_glr_plot_ehrenfest_irreversibility_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/ehrenfest_urn/plot_ehrenfest_irreversibility`

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      <div class="sphx-glr-thumbnail-title">The Ehrenfest urn: reversibility, recurrence, and the arrow of time</div>
    </div>


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    </div>

Interactive Plotly visualizations
-----------------------------------

Pan/zoom/hover-enabled counterparts of the static Matplotlib plots.


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    <div class="sphx-glr-thumbcontainer" tooltip="Every other visualizer in physicskit.statphys is Matplotlib-based: static, ideal for publication figures and this documentation&#x27;s gallery. physicskit.statphys.visualizers.interactive renders the same information as pan/zoom/hover-enabled Plotly figures instead, convenient for interactively exploring a parameter sweep or a dense vector field in a Jupyter notebook. This example builds one such figure for each of three lattice/particle models used throughout this gallery -- the 2D Ising ferromagnet, the 2D XY model, and a Lennard-Jones gas -- and saves each as a standalone HTML file.">

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  .. image:: /api/gallery/statphys/interactive/images/thumb/sphx_glr_plot_interactive_dashboards_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/interactive/plot_interactive_dashboards`

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      <div class="sphx-glr-thumbnail-title">Interactive Plotly visualizations</div>
    </div>


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    </div>

Ising model
-----------

The 2D ferromagnetic Ising model: Onsager's exactly solved second-order phase
transition.


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    <div class="sphx-glr-thumbcontainer" tooltip="The 2D Ising model places a spin s_i = \pm 1 on every site of a periodic L \times L lattice, with Hamiltonian">

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  .. image:: /api/gallery/statphys/ising/images/thumb/sphx_glr_plot_critical_slowing_down_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/ising/plot_critical_slowing_down`

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      <div class="sphx-glr-thumbnail-title">Critical slowing down: Metropolis vs. the Wolff cluster algorithm</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A single finite lattice can only show a rounded phase transition -- true singularities require an infinite system. Finite-size scaling theory turns this apparent limitation into a tool: by simulating several lattice sizes and comparing how observables scale with L, the exponents of the infinite system can be extracted without ever simulating one.">

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  .. image:: /api/gallery/statphys/ising/images/thumb/sphx_glr_plot_finite_size_scaling_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/ising/plot_finite_size_scaling`

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      <div class="sphx-glr-thumbnail-title">Finite-size scaling: extracting critical exponents</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Away from T_C, the 2D Ising lattice&#x27;s equilibrium configurations look qualitatively different above and below the transition: a fine, short-range-correlated speckle in the disordered phase, or a few large, slowly fluctuating domains in the ordered phase. Exactly at T_C, neither picture applies -- the correlation length">

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  .. image:: /api/gallery/statphys/ising/images/thumb/sphx_glr_plot_ising_criticality_animation_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/ising/plot_ising_criticality_animation`

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      <div class="sphx-glr-thumbnail-title">Watching criticality: the Ising lattice at T_C</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="The 2D Ising model places a spin s_i = \pm 1 on every site of a periodic L \times L lattice, with Hamiltonian">

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  .. image:: /api/gallery/statphys/ising/images/thumb/sphx_glr_plot_ising_phase_transition_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/ising/plot_ising_phase_transition`

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      <div class="sphx-glr-thumbnail-title">Onsager's exact solution: the 2D Ising phase transition</div>
    </div>


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    </div>

Interface growth
-----------------

The Kardar-Parisi-Zhang universality class via Restricted Solid-On-Solid growth.


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    <div class="sphx-glr-thumbcontainer" tooltip="Kardar, Parisi, and Zhang&#x27;s 1986 equation for a growing interface,">

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  .. image:: /api/gallery/statphys/kpz/images/thumb/sphx_glr_plot_kpz_roughening_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/kpz/plot_kpz_roughening`

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      <div class="sphx-glr-thumbnail-title">Universal interface roughening: the Kardar-Parisi-Zhang universality class</div>
    </div>


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Landau mean-field theory
-------------------------

Order parameters, broken symmetry, and critical exponents from a free-energy expansion alone.


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    <div class="sphx-glr-thumbcontainer" tooltip="Rather than derive a phase transition from a microscopic Hamiltonian, Landau (1937) asked what the free energy must look like near a continuous transition, given only the symmetry of an order parameter m that the disordered phase forbids from appearing at odd powers:">

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  .. image:: /api/gallery/statphys/landau_theory/images/thumb/sphx_glr_plot_landau_mean_field_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/landau_theory/plot_landau_mean_field`

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      <div class="sphx-glr-thumbnail-title">Landau mean-field theory: order parameters from symmetry alone</div>
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Molecular dynamics
------------------

The Lennard-Jones gas, Velocity Verlet integration, and Boltzmann's
H-theorem.


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    <div class="sphx-glr-thumbcontainer" tooltip="This example simulates a 2D periodic gas of N=200 particles interacting pairwise through the Lennard-Jones potential">

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  .. image:: /api/gallery/statphys/molecular_dynamics/images/thumb/sphx_glr_plot_maxwell_boltzmann_relaxation_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/molecular_dynamics/plot_maxwell_boltzmann_relaxation`

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      <div class="sphx-glr-thumbnail-title">Boltzmann's H-theorem: relaxation to the Maxwell-Boltzmann distribution</div>
    </div>


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Nonequilibrium work relations
-------------------------------

Jarzynski's equality: exact equilibrium free energies from irreversible work measurements.


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    <div class="sphx-glr-thumbcontainer" tooltip="The second law only bounds the average work needed to drive a system between two equilibrium states, \langle W \rangle \ge \Delta F, with equality only in the reversible, infinitely slow limit. Jarzynski (1997) found an exact equality hiding behind that inequality:">

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  .. image:: /api/gallery/statphys/nonequilibrium_work/images/thumb/sphx_glr_plot_jarzynski_equality_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/nonequilibrium_work/plot_jarzynski_equality`

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      <div class="sphx-glr-thumbnail-title">Jarzynski's equality: exact free energies from irreversible work</div>
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Percolation
-----------

Site and bond percolation: a purely geometric critical phenomenon.


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    <div class="sphx-glr-thumbcontainer" tooltip="Percolation theory shows that a sharp, universal phase transition does not require energy, temperature, or even dynamics -- pure geometry suffices. As the occupation probability p of a random lattice increases past a critical threshold p_c, an infinite spanning cluster suddenly appears. This example uses Hoshen-Kopelman cluster labeling to estimate the spanning probability curve P_{\text{span}}(p) for site percolation on the square lattice (p_c \approx 0.593), and visualizes the fractal, self-similar spanning cluster right at criticality.">

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  .. image:: /api/gallery/statphys/percolation/images/thumb/sphx_glr_plot_percolation_threshold_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/percolation/plot_percolation_threshold`

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      <div class="sphx-glr-thumbnail-title">Percolation: a purely geometric phase transition</div>
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    </div>

Potts model
-----------

The q-state generalization of the Ising model, contrasting continuous and
discontinuous transitions.


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    <div class="sphx-glr-thumbcontainer" tooltip="The Potts model generalizes the Ising model (q=2) to q discrete states per site: each site of a periodic L \times L lattice carries a state s_i \in \{0, \ldots, q-1\}, and the Hamiltonian rewards neighboring sites for matching,">

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  .. image:: /api/gallery/statphys/potts_model/images/thumb/sphx_glr_plot_potts_order_of_transition_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/potts_model/plot_potts_order_of_transition`

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      <div class="sphx-glr-thumbnail-title">First- versus second-order transitions in the q-state Potts model</div>
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Quantum statistics
-------------------

Bose-Einstein and Fermi-Dirac occupation, and Bose-Einstein condensation.


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    <div class="sphx-glr-thumbcontainer" tooltip="Classical Maxwell-Boltzmann statistics assumes particles are distinguishable and unrestricted in how they can share states. Quantum mechanics changes this in two opposite ways depending on particle spin. Bosons (integer spin) obey the Bose-Einstein distribution">

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  .. image:: /api/gallery/statphys/quantum_statistics/images/thumb/sphx_glr_plot_bose_fermi_distributions_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/quantum_statistics/plot_bose_fermi_distributions`

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      <div class="sphx-glr-thumbnail-title">Quantum statistics: Bose-Einstein, Fermi-Dirac, and condensation</div>
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Random walks
------------

Diffusion, the Einstein relation, and the central limit theorem.


.. raw:: html

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    <div class="sphx-glr-thumbcontainer" tooltip="Einstein&#x27;s 1905 analysis of Brownian motion related a microscopically invisible quantity -- the erratic jostling of a pollen grain by individual water molecules -- to a macroscopically measurable one, the diffusion coefficient D, via \langle r^2(t) \rangle = 2 d D t in d spatial dimensions. This was the first quantitative link between atomic-scale motion and observable matter, and it played a central role in convincing skeptics that atoms were real. This example simulates an ensemble of 500 independent walkers on a 2D lattice, each taking unit steps along a randomly chosen axis at every time step, recovers D from their mean squared displacement, and separately shows the central limit theorem in action: however non-Gaussian the individual steps are, the distribution of the walkers&#x27; displacement after many steps is Gaussian.">

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  .. image:: /api/gallery/statphys/random_walk/images/thumb/sphx_glr_plot_diffusion_and_clt_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/random_walk/plot_diffusion_and_clt`

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      <div class="sphx-glr-thumbnail-title">Diffusion, the Einstein relation, and the central limit theorem</div>
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Renormalization group
----------------------

Kadanoff block-spin coarse-graining and RG fixed points.


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    <div class="sphx-glr-thumbcontainer" tooltip="Real-space renormalization group (RG) transformations reveal why critical phenomena are universal. Starting from an Ising configuration (H = -J \sum_{\langle i,j \rangle} s_i s_j, s_i = \pm 1) equilibrated at temperature T, Kadanoff&#x27;s construction repeatedly replaces each non-overlapping 2\times2 block of spins by a single effective spin via majority rule,">

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  .. image:: /api/gallery/statphys/renormalization/images/thumb/sphx_glr_plot_block_spin_rg_flow_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/renormalization/plot_block_spin_rg_flow`

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      <div class="sphx-glr-thumbnail-title">Kadanoff block-spin renormalization group flow</div>
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Self-organized criticality
--------------------------

The Bak-Tang-Wiesenfeld sandpile: power laws without fine-tuning.


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    <div class="sphx-glr-thumbcontainer" tooltip="Ordinary critical phenomena (the Ising transition, percolation) require tuning a parameter -- temperature, occupation probability -- to a precise critical value. Bak, Tang, and Wiesenfeld&#x27;s 1987 sandpile model showed that some driven, dissipative systems tune themselves to criticality through their own dynamics, with no external parameter to adjust.">

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  .. image:: /api/gallery/statphys/sandpile/images/thumb/sphx_glr_plot_btw_avalanches_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/sandpile/plot_btw_avalanches`

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      <div class="sphx-glr-thumbnail-title">Self-organized criticality: the Bak-Tang-Wiesenfeld sandpile</div>
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Spin glasses
------------

The Edwards-Anderson Ising spin glass: quenched disorder and frustration.


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    <div class="sphx-glr-thumbcontainer" tooltip="A ferromagnet&#x27;s bonds all agree on which alignment lowers energy. A spin glass&#x27;s bonds are quenched random variables -- some ferromagnetic, some antiferromagnetic -- so a generic plaquette is frustrated: no spin configuration can satisfy all four of its bonds simultaneously. The Edwards-Anderson model uses the same Ising Hamiltonian form as a ferromagnet, but with random bond couplings on a periodic L \times L lattice:">

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  .. image:: /api/gallery/statphys/spin_glass/images/thumb/sphx_glr_plot_edwards_anderson_frustration_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/spin_glass/plot_edwards_anderson_frustration`

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      <div class="sphx-glr-thumbnail-title">Spin glasses: frustration and the Edwards-Anderson order parameter</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Where the Edwards-Anderson model puts random +/-J bonds on a lattice, the Sherrington-Kirkpatrick (SK) model in 1975 went to the opposite extreme: every spin couples to every other spin, with independent Gaussian bonds">

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  .. image:: /api/gallery/statphys/spin_glass/images/thumb/sphx_glr_plot_sk_overlap_distribution_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/spin_glass/plot_sk_overlap_distribution`

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      <div class="sphx-glr-thumbnail-title">The Sherrington-Kirkpatrick model and Parisi's replica overlap</div>
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XY model
--------

The 2D classical XY model and the topological Kosterlitz-Thouless
transition.


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    <div class="sphx-glr-thumbcontainer" tooltip="The 2D XY model places a planar spin angle \theta_i \in [0,2\pi) on every site of a periodic lattice, with Hamiltonian">

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  .. image:: /api/gallery/statphys/xy_model/images/thumb/sphx_glr_plot_kt_vortex_unbinding_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/xy_model/plot_kt_vortex_unbinding`

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      <div class="sphx-glr-thumbnail-title">The Kosterlitz-Thouless transition: vortex-antivortex unbinding</div>
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Partition function zeros
-------------------------

The Lee-Yang circle theorem: phase transitions from the complex-fugacity structure of Z.


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    <div class="sphx-glr-thumbcontainer" tooltip="A finite system&#x27;s partition function is a finite sum of manifestly analytic terms -- so where can a genuine phase transition, a true discontinuity in the free energy, possibly come from? Lee and Yang&#x27;s 1952 answer: continue the partition function into the complex fugacity plane and look at its zeros. Writing the Ising Hamiltonian with a field,">

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  .. image:: /api/gallery/statphys/yang_lee/images/thumb/sphx_glr_plot_yang_lee_zeros_thumb.png
    :alt:

  :doc:`/api/gallery/statphys/yang_lee/plot_yang_lee_zeros`

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      <div class="sphx-glr-thumbnail-title">The Lee-Yang circle theorem: partition function zeros in the complex plane</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/statphys/ehrenfest_urn/index.rst
   /api/gallery/statphys/interactive/index.rst
   /api/gallery/statphys/ising/index.rst
   /api/gallery/statphys/kpz/index.rst
   /api/gallery/statphys/landau_theory/index.rst
   /api/gallery/statphys/molecular_dynamics/index.rst
   /api/gallery/statphys/nonequilibrium_work/index.rst
   /api/gallery/statphys/percolation/index.rst
   /api/gallery/statphys/potts_model/index.rst
   /api/gallery/statphys/quantum_statistics/index.rst
   /api/gallery/statphys/random_walk/index.rst
   /api/gallery/statphys/renormalization/index.rst
   /api/gallery/statphys/sandpile/index.rst
   /api/gallery/statphys/spin_glass/index.rst
   /api/gallery/statphys/xy_model/index.rst
   /api/gallery/statphys/yang_lee/index.rst



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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_