# Glauber dynamics

**Glauber dynamics** is a [Markov chain](https://www.edgepedia.org/markov-chain) on the configurations of a spin system, introduced by [Roy J. Glauber](https://www.edgepedia.org/roy-j-glauber) in 1963, in which single spins are updated one at a time with transition rates chosen so that the chain has the Boltzmann (Gibbs) distribution of the [Ising model](https://www.edgepedia.org/ising-model) as its unique stationary distribution<sup>[1](https://cir.nii.ac.jp/crid/1364233270594518016)</sup>. It serves two roles at once: a physical model of how a magnet relaxes toward equilibrium in contact with a heat bath, and an algorithm (equivalent to Gibbs sampling or heat-bath dynamics) for sampling from distributions whose normalization constant cannot be computed directly<sup>[2](https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/Ising.pdf)</sup><sup> • </sup><sup>[3](https://www.math.cmu.edu/~gautam/c/2025-326/notes/sampling4-mcmc.html)</sup>.

| Key fact | Detail |
|---|---|
| Origin | Roy J. Glauber, "Time-Dependent Statistics of the Ising Model", *Journal of Mathematical Physics* 4(2), 294–307, published 1 February 1963<sup>[1](https://cir.nii.ac.jp/crid/1364233270594518016)</sup> |
| Update rule | A spin flips at rate \( w = 1/(1 + e^{\beta \Delta E}) \), where \( \Delta E \) is the energy change of the flip and \( \beta \) the inverse temperature<sup>[4](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)</sup> |
| Stationary law | Detailed balance holds, so the unique stationary distribution is the Boltzmann distribution \( \pi(s) \propto e^{-\beta H(s)} \)<sup>[4](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)</sup><sup> • </sup><sup>[3](https://www.math.cmu.edu/~gautam/c/2025-326/notes/sampling4-mcmc.html)</sup>  |
| Sampling without \( Z \) | The chain samples from \( \pi \) without knowing the partition function \( Z \), provided one-site conditional marginals are easy to draw from<sup>[3](https://www.math.cmu.edu/~gautam/c/2025-326/notes/sampling4-mcmc.html)</sup> |
| Mean-field mixing | Curie-Weiss Ising: \( \Theta(n \log n) \) for \( \beta < 1 \), \( \Theta(n^{3/2}) \) at \( \beta = 1 \), exponential in \( n \) for \( \beta > 1 \)<sup>[5](https://arxiv.org/abs/0806.1906)</sup> |
| Critical 2D Ising | The inverse spectral gap at \( \beta_c = \tfrac{1}{2}\log(1+\sqrt{2}) \) is polynomial in the side length, the first rigorous polynomial bound (Lubetzky and Sly, 2010)<sup>[6](https://arxiv.org/abs/1001.1613)</sup> |
| Conserving counterpart | Kawasaki dynamics, the spin-exchange version that conserves magnetization, was constructed after Glauber's single-spin-flip model<sup>[7](https://www.cambridge.org/core/books/abs/nonequilibrium-statistical-mechanics-in-one-dimension/kinetic-ising-models-with-competing-dynamics-mappings-correlations-steady-states-and-phase-transitions/1B3A5B1B7F10A0BAF255EE6A8FA78E2B)</sup> |

## Definition and update rule

The Ising model is an equilibrium model: it assigns probabilities to spin configurations but says nothing about how they change in time. Glauber's construction couples the spins to an external agency, a heat reservoir, which causes them to change state randomly, with the transition probabilities for any one spin depending on the values of its neighbors<sup>[1](https://cir.nii.ac.jp/crid/1364233270594518016)</sup>.

In the single-spin-flip version, only transitions that flip one spin at a time are allowed. When the current configuration is \( s \) and flipping the spin at site \( i \) gives \( s_i \), the flip rate is<sup>[4](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)</sup>

\[ w(s_i \to -s_i) = \frac{1}{1 + e^{\beta \Delta E}}, \qquad \Delta E = H(s_i) - H(s). \]

This heat-bath rate satisfies detailed balance, \( \pi(s)\, w(s \to s_i) = \pi(s_i)\, w(s_i \to s) \), which forces the stationary distribution to be the [Boltzmann distribution](https://www.edgepedia.org/boltzmann-distribution) \( \pi(s) \propto e^{-\beta H(s)} \)<sup>[4](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)</sup>. The rate has a physical reading: each spin instantaneously equilibrates with its local mean field at unit rate, so magnetization traces, time correlations, and relaxation times computed from a Glauber trajectory carry dynamical content, not just equilibrium statistics<sup>[4](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)</sup>.

**Acceptance probability.** In discrete-time [Monte Carlo](https://www.edgepedia.org/monte-carlo) form, Glauber's rule is an acceptance probability \( p = 1/(1 + e^{x}) \) with \( x = \beta \Delta E \), compared with the [Metropolis](https://www.edgepedia.org/metropolis) choice \( p = \min\{1, e^{-x}\} \)<sup>[8](https://citeseerx.ist.psu.edu/document?doi=74a2de6b9eae00af33770736fa37a8dcb1020cb7&repid=rep1&type=pdf)</sup>. Both satisfy detailed balance. The difference shows at zero energy change: Metropolis always accepts \( \Delta E = 0 \) moves, while the Glauber rule accepts them with probability \( 1/2 \)<sup>[9](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_notes.pdf)</sup>.

## Relation to Gibbs sampling and Metropolis–Hastings

The same chain reappears in statistics and computer science under other names. As the Gibbs sampler or heat-bath dynamics, a configuration is updated by choosing a vertex \( v \) uniformly at random and assigning it a spin drawn from the Gibbs distribution conditional on all other spins; all other spins are left unchanged<sup>[2](https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/Ising.pdf)</sup>. This works without computing the partition function \( Z \), because only conditional one-site marginals are needed, and the resulting chain is irreducible, aperiodic, and reversible with stationary distribution \( \pi \)<sup>[3](https://www.math.cmu.edu/~gautam/c/2025-326/notes/sampling4-mcmc.html)</sup>.

The Glauber chain is also exactly a Metropolis–Hastings chain whose proposal mechanism proposes resampling one site from its conditional distribution, so that the acceptance probability of every proposed move is 1<sup>[10](https://www.math.cmu.edu/~gautam/c/2024-387/notes/09-metropolis-hastings2.html)</sup>. In a continuous-time version, each site holds an independent rate-1 Poisson clock, and when a clock rings the site's spin is redrawn from its local conditional distribution; this idealized parallelization was noted as early as Glauber's original work<sup>[11](https://www.researchgate.net/publication/356027037_Simple_Parallel_Algorithms_for_Single-Site_Dynamics)</sup>.

## Mixing times

How fast the chain approaches equilibrium is the central quantitative question, and the answer depends strongly on temperature and graph structure.

**High temperature.** For an \( n \)-vertex graph of maximal degree \( \Delta \), if \( \Delta \tanh(\beta) < 1 \) the Ising Glauber dynamics mixes in \( O\!\left( \frac{n}{1 - \Delta \tanh(\beta)} \log n \right) \) steps<sup>[12](https://www.math.ubc.ca/~jhermon/Mixing/Coupling_from_the_past.pdf)</sup>. The same chain is monotone with respect to the pointwise ordering of spin configurations, which supports coupling arguments used throughout the mixing literature<sup>[12](https://www.math.ubc.ca/~jhermon/Mixing/Coupling_from_the_past.pdf)</sup>.

**Mean-field behavior.** On the complete graph (the Curie-Weiss model), the mixing time is \( \Theta(n \log n) \) for \( \beta < 1 \), with a cutoff at time \( [2(1-\beta)]^{-1} n \log n \); it is \( \Theta(n^{3/2}) \) at the critical point \( \beta = 1 \); and it is exponential in \( n \) for \( \beta > 1 \)<sup>[5](https://arxiv.org/abs/0806.1906)</sup>. Ding, Lubetzky, and Peres showed that a scaling window of order \( 1/\sqrt{n} \) around \( \beta_c = 1 \) connects these regimes: for \( \beta = 1 - \delta \) with \( \delta^2 n \to \infty \), the mixing time has order \( (n/\delta) \log(\delta^2 n) \) with cutoff, while for \( \beta = 1 + \delta \) it has order \( (n/\delta) \exp\!\left( (\tfrac{3}{4} + o(1)) \delta^2 n \right) \) without cutoff<sup>[5](https://arxiv.org/abs/0806.1906)</sup>. The exponential low-temperature cost comes from the magnetization chain having two centers of mass at \( \pm \zeta n \), separated by an exponentially long commute time<sup>[5](https://arxiv.org/abs/0806.1906)</sup>.

**Critical slowdown in two dimensions.** Physicists' picture of the phase transition predicts that at high temperature the inverse spectral gap is \( O(1) \), at the critical \( \beta_c \) it is polynomial in the side length, and at low temperature it is exponential in it<sup>[6](https://arxiv.org/abs/1001.1613)</sup>. Lubetzky and Sly established the first rigorous polynomial upper bound at criticality for the square lattice, showing that on a finite box with arbitrary boundary conditions the inverse gap at \( \beta_c = \tfrac{1}{2}\log(1+\sqrt{2}) \) is at most \( n^{C} \) for an absolute constant \( C \), confirming the critical slowdown for the Ising model on \( \mathbb{Z}^2 \)<sup>[6](https://arxiv.org/abs/1001.1613)</sup>.

**Initialization matters at low temperature.** Worst-case low-temperature mixing is exponential because metastable phases are separated by narrow bottlenecks. For the dynamics on an N-site torus in Z^d, Gheissari and Sinclair established that when the initial state is an even blend of the all-plus and all-minus ground states, the mixing time grows as N^{1+o(1)} when d = 2 and stays bounded by a quasi-polynomial for every d ≥ 3, throughout the subcritical temperature regime<sup>[13](https://arxiv.org/abs/2106.11296)</sup>. On random regular graphs at sufficiently low temperature, the same initialization gives optimal \( O(N \log N) \) mixing<sup>[13](https://arxiv.org/abs/2106.11296)</sup>.

## Related dynamics and extensions

Glauber's single-spin-flip model was the first dynamical generalization of the Ising model, built to describe relaxation toward equilibrium; Kawasaki then constructed a spin-exchange version in which pairs of spins are swapped, so that magnetization is conserved<sup>[7](https://www.cambridge.org/core/books/abs/nonequilibrium-statistical-mechanics-in-one-dimension/kinetic-ising-models-with-competing-dynamics-mappings-correlations-steady-states-and-phase-transitions/1B3A5B1B7F10A0BAF255EE6A8FA78E2B)</sup>. Combining Glauber flips and Kawasaki exchanges preserves detailed balance when both satisfy it with respect to the same distribution; driven combinations can instead break detailed balance and produce nonequilibrium steady states whose critical exponents have been compared with those of the equilibrium Ising model<sup>[14](https://www.scielo.br/j/bjp/a/SsMm94PpKKS7FztDjMqGkkm/?lang=en)</sup><sup> • </sup><sup>[17](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.5802)</sup>.

The general definition extends beyond the Ising model: given an energy function \( H(s) \), an inverse temperature \( \beta \ge 0 \), and a nonnegative symmetric transition rate matrix, Glauber dynamics is the continuous-time Markov chain satisfying detailed balance with rates \( \gamma_{s's} = 1/(1 + e^{\beta (H(s') - H(s))}) \); it can be simulated efficiently whenever the energy differences \( H(s') - H(s) \) can be computed efficiently, as in classical Ising models where local interactions make the energy difference of a few-spin flip a local computation<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC12527926/)</sup>. The same construction applies to Potts models with \( q \ge 3 \) states, where the mean-field Glauber dynamics undergoes a critical slowdown at an inverse temperature \( \beta_s(q) \) strictly lower than the thermodynamic critical point \( \beta_c(q) \)<sup>[16](https://arxiv.org/abs/1204.4503)</sup>.

## References

1. [Roy J. Glauber (1963). Time-Dependent Statistics of the Ising Model. Journal of Mathematical Physics 4(2), 294–307.](https://cir.nii.ac.jp/crid/1364233270594518016)
2. [Mossel, E., Sly, A. Exact Thresholds for Ising–Gibbs Samplers on General Graphs. Annals of Probability.](https://www.math.cmu.edu/~af1p/Teaching/MCC17/Papers/Ising.pdf)
3. [Markov Chain Monte Carlo lecture notes, Carnegie Mellon University (Gautam Iyer).](https://www.math.cmu.edu/~gautam/c/2025-326/notes/sampling4-mcmc.html)
4. [Glauber Dynamics: Time Evolution of Ising Spins, Statmech4ChemBio lecture notes.](https://dpotoyan.github.io/Statmech4ChemBio/glauber/)
5. [Ding, J., Lubetzky, E., Peres, Y. (2008). The mixing time evolution of Glauber dynamics for the mean-field Ising model. arXiv:0806.1906.](https://arxiv.org/abs/0806.1906)
6. [Lubetzky, E., Sly, A. (2010). Critical Ising on the square lattice mixes in polynomial time. arXiv:1001.1613.](https://arxiv.org/abs/1001.1613)
7. [Rácz, Z. Kinetic Ising models with competing dynamics. In Nonequilibrium Statistical Mechanics in One Dimension, Cambridge University Press.](https://www.cambridge.org/core/books/abs/nonequilibrium-statistical-mechanics-in-one-dimension/kinetic-ising-models-with-competing-dynamics-mappings-correlations-steady-states-and-phase-transitions/1B3A5B1B7F10A0BAF255EE6A8FA78E2B)
8. [An Introduction to Monte Carlo Methods for Statistical Physics (Glauber acceptance probability).](https://citeseerx.ist.psu.edu/document?doi=74a2de6b9eae00af33770736fa37a8dcb1020cb7&repid=rep1&type=pdf)
9. [Computational Statistical Physics lecture notes, ETH Zürich (Glauber/heat-bath dynamics).](https://ethz.ch/content/dam/ethz/special-interest/phys/theoretical-physics/itp-dam/documents/compstatphys/CompStatPhysPartI_notes.pdf)
10. [Metropolis Hastings Revisited, Carnegie Mellon University lecture notes.](https://www.math.cmu.edu/~gautam/c/2024-387/notes/09-metropolis-hastings2.html)
11. [Simple Parallel Algorithms for Single-Site Dynamics.](https://www.researchgate.net/publication/356027037_Simple_Parallel_Algorithms_for_Single-Site_Dynamics)
12. [Coupling from the past (mixing of monotone Glauber dynamics), University of British Columbia lecture notes.](https://www.math.ubc.ca/~jhermon/Mixing/Coupling_from_the_past.pdf)
13. [Gheissari, R., Sinclair, A. (2021). Low-temperature Ising dynamics with random initializations. arXiv:2106.11296.](https://arxiv.org/abs/2106.11296)
14. [Figueiredo, W., Grandi, B.C.S. Competing spin dynamics in Ising systems. Brazilian Journal of Physics.](https://www.scielo.br/j/bjp/a/SsMm94PpKKS7FztDjMqGkkm/?lang=en)
15. [Efficient quantum thermal simulation. PMC.](https://pmc.ncbi.nlm.nih.gov/articles/PMC12527926/)
16. [Glauber Dynamics for the Mean-Field Potts Model. arXiv:1204.4503.](https://arxiv.org/abs/1204.4503)
17. [journals.aps.org](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.40.5802)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Physics methods, practice, and community › Applied and interdisciplinary physics › Computational and simulation physics › Monte Carlo methods in physics › Markov chain Monte Carlo in physics*

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