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| Curie–Weiss model | |
|---|---|
| Name | Curie–Weiss model |
| Researcher | Pierre Curie, Pierre Weiss |
| Field | Statistical mechanics, Condensed matter physics |
| Introduced | 1907 |
| Notable for | Mean-field description of ferromagnetism |
Curie–Weiss model The Curie–Weiss model is a mean-field lattice model of ferromagnetism introduced by Pierre Curie and Pierre Weiss that captures collective magnetic ordering via long-range interactions; it provides a paradigmatic example of a phase transition and spontaneous symmetry breaking used in works by Ludwig Boltzmann, James Clerk Maxwell, and later formalized in studies by Lev Landau, Lars Onsager, and Rudolf Peierls. The model's simplicity made it influential for researchers such as John von Neumann, Emil Artin, and Andrey Kolmogorov in developing statistical and mathematical techniques later applied in fields by Richard Feynman, Enrico Fermi, and Paul Dirac.
The Curie–Weiss model is defined on a set of N spins with energy determined by an all-to-all coupling and external field, a formulation that parallels approaches by Pierre Curie and Pierre Weiss and is analogous in spirit to models studied by Isidor Isaac Rabi and Ernest Rutherford. The Hamiltonian H_N(s) = - (J/2N) (sum_i s_i)^2 - h sum_i s_i uses spin variables s_i ∈ {±1} and parameters J and h, a structure resonant with treatments by Max Planck, Albert Einstein, and Niels Bohr in statistical ensembles. The canonical ensemble for the model employs a partition function Z_N(β,h) summing over configurations, an approach central to analyses by Josiah Willard Gibbs, Paul Langevin, and Ludwig Boltzmann.
The thermodynamic limit N→∞ yields free energy per spin and prediction of a ferromagnetic phase for J>0 below a critical temperature T_c, results discussed alongside classical expositions by Lev Landau, Landau and experimental context from Pierre Curie and Heike Kamerlingh Onnes. Specific heat, internal energy, and magnetization follow from derivatives of the free energy, a methodology used in studies by Wilhelm Lenz, Erwin Schrödinger, and Wolfgang Pauli. The emergence of spontaneous magnetization for h→0 and T
The Curie–Weiss model is the mean-field or infinite-range analog of the nearest-neighbor Ising model originally studied on lattices by Wilhelm Lenz, Ernst Ising, and later solved in two dimensions by Lars Onsager; mean-field approximations trace conceptually to techniques by Pierre Weiss, Lev Landau, and Pierre-Gilles de Gennes in later contexts. One derives the self-consistency equation for magnetization by replacing local fields with their average, an approximation used across work by Enrico Fermi, Richard Feynman, and Julian Schwinger. Connections to variational principles and Gibbs measures echo mathematical frameworks from Andrey Kolmogorov, Alfréd Rényi, and André Weil.
Solving the Curie–Weiss model yields the self-consistent equation m = tanh(β(J m + h)), a relation appearing in treatments by Pierre Weiss and later texts by L. D. Landau, Kadanoff, and Kenneth Wilson. The magnetic susceptibility χ = ∂m/∂h shows a Curie–Weiss law χ ∝ |T-T_c|^{-1} above T_c, a behavior first identified experimentally by Pierre Curie and theoretically by Pierre Weiss and analyzed in statistical expositions by Josiah Willard Gibbs, Rudolf Peierls, and John Hubbard. Mean-field critical amplitudes and temperature dependence were incorporated into renormalization reasoning by Kenneth Wilson and Michael Fisher.
Near T_c the Curie–Weiss model exhibits critical exponents β = 1/2, γ = 1, δ = 3, and α = 0 (logarithmic corrections absent in mean-field), values that fit the mean-field universality class discussed in works by Kenneth Wilson, Michael Fisher, Leo Kadanoff, and Benjamin Widom. The model's universality links to systems studied by Evgeny Lifshitz, Isaac Newton, and Pierre Curie where long-range order and symmetry breaking are central; renormalization group insights from Kenneth Wilson contextualize why Curie–Weiss exponents differ from low-dimensional results found by Lars Onsager. Applications in critical phenomena have been referenced in treatments by John Cardy, Bernard Derrida, and Giovanni Jona-Lasinio.
Generalizations include q-state Potts mean-field variants popularized by R. B. Potts and Franz Wegner, continuous-spin spherical models introduced by T. H. Berlin and M. Kac, random-field Curie–Weiss studied in works by Mézard, Giorgio Parisi, and Marc Mézard, and diluted or inhomogeneous versions linked to models by Elliott Lieb and Barry McCoy. Connections to spin glass theory reference pioneering contributions by David Sherrington, Scott Kirkpatrick, and Giorgio Parisi, while analogs appear in social science models influenced by Thomas Schelling, Martha Nussbaum, and Robert Axelrod.
Rigorous analysis of the Curie–Weiss model's thermodynamic limit, fluctuation theorems, large deviations, and metastability has been developed by mathematicians such as Olivier Lanford, David Ruelle, Herbert Spohn, Fritz Haake, Michael Aizenman, and Roberto Fernández. Results include concentration of measure and central limit theorems for magnetization proven in work by Stanislav Smirnov, Alain-Sol Sznitman, and Giambattista Giacomin, and rigorous characterization of phase coexistence and metastability contributed by Olav Kallenberg, H.-O. Georgii, and S. B. Shlosman. Connections to combinatorial and probabilistic limits appear in analyses by Paul Erdős, Alfréd Rényi, and Boris Pittel.
Category:Statistical mechanics models