Brownian Riesz Gas Dynamics

• Brownian Riesz gas is a class of stochastic many-body systems characterized by overdamped diffusion coupled with singular Riesz and logarithmic repulsive interactions.
• The formulation unifies diverse models such as the Dyson log gas, circular Riesz gases, and quantum vortex gases, providing a common framework for equilibrium and fluctuation analysis.
• Research in this area explores mean-field limits, scaling laws, and phase transitions under various confinements, offering precise insights into dynamic and equilibrium behaviors.

Searching arXiv for the cited papers and closely related Brownian/Riesz gas works to ground the article. arXiv Search Query: "all:Brownian Riesz gas OR all:Riesz gas Brownian motion Dyson Coulomb vortices" Brownian Riesz gas denotes a class of stochastic many-body systems in which Brownian motion, or an overdamped diffusion, is coupled to singular repulsive interactions of Riesz type $|x-y|^{-s}$, together with the logarithmic case $-\log|x-y|$. In the current literature this includes one-dimensional systems of ordered Brownian particles with $V'(x)=-x/|x|^{\alpha+1}$, the Dyson log gas as the case $\alpha=1$, circular Riesz gases on the torus, and two-dimensional quantum vortex gases whose effective interaction is logarithmic and becomes a Dyson-type Coulomb gas in the overdamped limit (Guillin et al., 2022, Boursier, 2022, Kuratsuji, 2022). The subject also includes the variational theory of the associated Gibbs measures, infinite-volume and propagation-of-chaos limits, and fluctuation laws for currents, gaps, and tagged particles (Byun et al., 3 Feb 2026, Suzuki, 2022, Dandekar et al., 2022).

1. Model classes and defining equations

A canonical one-dimensional Brownian Riesz gas is the system

$$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$

with $U$ a confining potential and

$$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$

Thus

$V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$

The case $\alpha=1$, $U(x)=\lambda x^2/2$, and $-\log|x-y|$0 is the generalized Dyson Brownian motion, so the Brownian log gas is the $-\log|x-y|$1 member of the Brownian Riesz gas family (Guillin et al., 2022).

On the circle $-\log|x-y|$2, the equilibrium Riesz gas is defined by the Hamiltonian

$-\log|x-y|$3

where $-\log|x-y|$4 is the periodic Riesz kernel and the Gibbs measure is

$-\log|x-y|$5

In this setting $-\log|x-y|$6 is the long-range regime, while $-\log|x-y|$7 is short-range or hypersingular (Boursier, 2022).

A structurally different realization appears in two-dimensional quantum condensates. Starting from the time-dependent Landau-Ginzburg theory and the vortex phase ansatz

$-\log|x-y|$8

the effective Hamiltonian for the vortex centers is

$-\log|x-y|$9

The interaction term is exactly the two-dimensional Coulomb, hence logarithmic, Riesz interaction, and the stationary density in the overdamped regime is the Gibbs law of a Dyson-type Coulomb gas (Kuratsuji, 2022).

A higher-dimensional equilibrium formulation uses

$V'(x)=-x/|x|^{\alpha+1}$0

with

$V'(x)=-x/|x|^{\alpha+1}$1

This gives the standard finite-$V'(x)=-x/|x|^{\alpha+1}$2 Riesz gas Hamiltonian in dimension $V'(x)=-x/|x|^{\alpha+1}$3, with Coulomb interaction at $V'(x)=-x/|x|^{\alpha+1}$4 and the logarithmic limit at $V'(x)=-x/|x|^{\alpha+1}$5 (Byun et al., 3 Feb 2026).

2. Microscopic derivations and stochastic generators

Brownian Riesz gases need not be postulated directly at the particle level. In the vortex construction, insertion of the vortex ansatz into the Landau-Ginzburg Lagrangian yields the canonical term

$V'(x)=-x/|x|^{\alpha+1}$6

so $V'(x)=-x/|x|^{\alpha+1}$7 form a canonical pair and satisfy

$V'(x)=-x/|x|^{\alpha+1}$8

After adding dissipation and Gaussian white noise,

$V'(x)=-x/|x|^{\alpha+1}$9

which can be rewritten as

$\alpha=1$0

The drift $\alpha=1$1 contains both a gradient part $\alpha=1$2 and a transverse Magnus part $\alpha=1$3, while the noise remains additive Gaussian white noise (Kuratsuji, 2022).

The corresponding generalized Fokker-Planck equation is

$\alpha=1$4

Because of the rotational drift, the generator is non-self-adjoint, or non-Hermitian in the Schrödinger representation. In the overdamping regime $\alpha=1$5, the transverse term becomes negligible and the equation reduces to the standard Smoluchowski form, with equilibrium

$\alpha=1$6

In the underdamping regime $\alpha=1$7, diffusion and gradient drift become perturbative and the dynamics reduces to a Liouville equation with a small collision term (Kuratsuji, 2022).

At infinite particle number, the modern configuration-space formulation is expressed by Dirichlet forms on the configuration space $\alpha=1$8. For $\alpha=1$9 and for the one-dimensional $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$0-circular Riesz gas with $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$1, a strongly local symmetric Dirichlet form is constructed whose symmetrising measure is the infinite-volume Gibbs law. The form satisfies the Bakry-Émery estimate $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$2, implies local Poincaré and local log-Sobolev inequalities, and yields a dual semigroup that is the unique $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$3-gradient flow of the Boltzmann-Shannon entropy (Suzuki, 2022).

3. Equilibrium measures, confinement, and variational structure

The equilibrium problem is encoded by the mean-field functional

$$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$4

whose minimizer $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$5 satisfies the Euler-Lagrange conditions

$$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$6

For rotationally symmetric higher-dimensional Riesz gases, one may prescribe a radial density on the unit ball,

$$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$7

and reconstruct an external field $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$8 for which $$dX^i_t=\sqrt{2\sigma_N}\,dB^i_t-U'(X^i_t)\,dt-\frac{1}{N}\sum_{j\neq i}V'(X^i_t-X^j_t)\,dt,\qquad i=1,\dots,N,$$9 is the equilibrium measure. Explicit families include densities proportional to $U$0 and equilibrium measures for purely power-type external fields, both written in terms of hypergeometric functions. The same work states that these $U$1 are the natural limiting stationary distributions for Brownian Riesz gases with radially symmetric confining fields (Byun et al., 3 Feb 2026).

One-dimensional trapping and boundary effects produce a sharper classification. For the harmonically confined Riesz gas with a hard wall at scaled position $U$2, the large-$U$3 equilibrium density exhibits three regimes. For $U$4, the scaled density is supported on $U$5, vanishes at the left edge, and approaches a nonzero constant at the wall. For $U$6, it is again supported on $U$7, vanishes at the left edge, and diverges algebraically at the wall with exponent $U$8. For $U$9, the density has an extended bulk part and a delta peak at the wall, separated by a hole; in this regime a first-order phase transition occurs when the wall crosses a critical value $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$0, and the amplitude of the delta peak jumps to $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$1 (Kethepalli et al., 2021).

Equilibrium fluctuation theory for trapped one-dimensional Riesz gases can be developed through linear statistics

$$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$2

For $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$3, the distribution of $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$4 has the large-deviation form

$$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$5

and the rate function exhibits an evaporation transition, where a large fluctuation of $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$6 is dominated by the largest $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$7. For $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$8, the higher cumulants of full counting statistics near the edge exhibit a distinct scaling regime as $$V'(x)=-\frac{x}{|x|^{\alpha+1}},\qquad \alpha\ge 1.$$9 (Doussal et al., 2024).

Potential-theoretic regularity of the background measure matters as well. For mean-field Coulomb and Riesz gases, the large deviation principle at $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$0 with rate functional continuous in $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$1 holds if and only if the prior measure is strongly determining; otherwise a zeroth-order phase transition at zero temperature may occur (Berman, 2018).

4. Mean-field limits, thermodynamic limit, and chaos

For the one-dimensional interacting diffusion with singular repulsive force $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$2, well-posedness depends on both the singularity and the microscopic noise. If $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$3, there is a unique strong solution starting from the ordered chamber $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$4, and particles never collide. If $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$5, the same remains true provided $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$6. Under quadratic confinement and $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$7, the empirical measure $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$8 converges to a deterministic curve $V(x)= \begin{cases} \dfrac{1}{\alpha-1}|x|^{-(\alpha-1)}, & \alpha>1,\[0.4em] -\log|x|, & \alpha=1. \end{cases}$9, and if $\alpha=1$0 then

$\alpha=1$1

uniform in time up to an exponentially decaying transient; for the logarithmic case $\alpha=1$2, this gives the rate $\alpha=1$3. The limiting McKean-Vlasov equation is identified in weak form, and $\alpha=1$4 emerges as the critical exponent for integrability (Guillin et al., 2022).

On the circle, the thermodynamic limit is controlled at microscopic scale. After rescaling so that the typical spacing is of order $\alpha=1$5, the finite-$\alpha=1$6 circular Riesz gas converges to a translation-invariant infinite-volume point process $\alpha=1$7. In the long-range regime $\alpha=1$8, the gap correlations decay as a power law with exponent $\alpha=1$9, and this decay is sharp up to $U(x)=\lambda x^2/2$0. The same process is hyperuniform, with

$U(x)=\lambda x^2/2$1

and the microscopic gap observables converge with a quantitative rate $U(x)=\lambda x^2/2$2 (Boursier, 2022).

These two directions—finite-$U(x)=\lambda x^2/2$3 propagation of chaos and infinite-volume Gibbs limits—show complementary aspects of Brownian Riesz gases. The first is a mean-field limit for ordered singular diffusions in one dimension; the second is a thermodynamic-limit description of microscopic equilibrium processes on the circle. This suggests that the expression “Brownian Riesz gas” functions as an umbrella notion rather than a single canonical model.

5. Dynamical fluctuations, currents, and tagged particles

For an infinite one-dimensional Brownian Riesz gas with pair potential

$U(x)=\lambda x^2/2$4

the overdamped Langevin dynamics leads to anomalous current and tagged-particle fluctuations. In the long-range regime $U(x)=\lambda x^2/2$5, the standard deviations of the integrated current and the tracer position grow as

$U(x)=\lambda x^2/2$6

while for $U(x)=\lambda x^2/2$7 the universal single-file exponent $U(x)=\lambda x^2/2$8 is recovered. The two-time correlations of the tagged-particle position have the same form as for fractional Brownian motion, and the marginal case $U(x)=\lambda x^2/2$9 is predicted to behave as $-\log|x-y|$00 at the level of the standard deviation (Dandekar et al., 2022).

A complementary weak-noise analysis on the circle studies fluctuations around the equally spaced crystal configuration. In the Brownian case, the exact space-time correlations are controlled by the dynamical exponent

$-\log|x-y|$01

and the gap roughness exponent

$-\log|x-y|$02

For $-\log|x-y|$03, the mean square displacement is sub-diffusive as $-\log|x-y|$04 for short-range interactions and $-\log|x-y|$05 for long-range interactions, and the amplitudes coincide with the recent macroscopic fluctuation theory predictions. The same formalism extends to run-and-tumble particles and shows that the active system crosses over to the Brownian Riesz gas beyond the persistence time $-\log|x-y|$06 (Touzo et al., 2024).

The static circular theory and the fluctuation theory are consistent at the level of long-range exponents. In the long-range regime, one finds both power-law spatial decay of equilibrium correlations with exponent $-\log|x-y|$07 and subdiffusive tagged-particle laws governed by the same nonlocal interaction parameter $-\log|x-y|$08. In the short-range regime, both the microscopic weak-noise theory and the fluctuating-hydrodynamic theory recover the standard single-file exponent.

6. Regimes, transitions, and conceptual scope

Several structural distinctions recur across the literature. One is the long-range versus short-range dichotomy: on the circle, $-\log|x-y|$09 yields non-summable long-range interactions and power-law gap correlations, whereas $-\log|x-y|$10 gives faster decay and effectively local diffusion (Boursier, 2022). Another is the overdamped versus underdamped distinction in the vortex gas, where the overdamped regime leads to a reversible Dyson-type equilibrium while the underdamped regime yields a transport equation with non-Hermitian generator (Kuratsuji, 2022).

A second recurrent theme is the role of confinement and background measure. Harmonic traps, power-type traps, hard walls, and conditioning on a prior all change the equilibrium problem qualitatively. The hard-wall problem for $-\log|x-y|$11 produces a first-order transition with a discontinuous wall condensate (Kethepalli et al., 2021). In mean-field Coulomb and Riesz gases, strong determinacy of the prior is equivalent to continuity of the zero-temperature limit and to the absence of a zeroth-order phase transition (Berman, 2018). In higher dimensions, explicit rotationally symmetric equilibrium densities can be reconstructed together with the confining field that generates them, and these densities serve as candidate stationary profiles for Brownian Riesz dynamics (Byun et al., 3 Feb 2026).

Taken together, these results indicate that Brownian Riesz gas is best understood as a research program centered on stochastic dynamics with singular repulsive Riesz interaction, rather than as a single model. Its current formulations include ordered diffusions in one dimension, Dyson-type logarithmic gases, infinite-volume Gibbs states on configuration space, Brownian vortex gases with gyrotropic drift, and trapped or wall-constrained equilibrium ensembles. What unifies them is the combination of Brownian noise or overdamped diffusion, Riesz or logarithmic repulsion, and a variational or generator-level description precise enough to resolve equilibrium measures, scaling limits, and fluctuation laws.