The Dean-Kawasaki equation and stochastic density functional theory

Pierre Illien

The Dean-Kawasaki equation and stochastic density functional theory

T0 review · 0 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The density of $N$ interacting Brownian particles obeys, without any approximation, a single stochastic partial differential equation—the Dean-Kawasaki equation—and this review shows how that exact equation links fluctuating…

desk verdict A careful, honest review of the Dean-Kawasaki equation that consolidates the field; no new results, but a reliable map for newcomers and a fair treatment of the mathematics. read the letter →

arxiv 2411.13467 v2 pith:IPFIAGFB submitted 2024-11-20 cond-mat.soft cond-mat.stat-mech

Pierre Illien This is my paper · ORCID

classification cond-mat.softcond-mat.stat-mech

keywords Dean-KawasakiequationstochasticdensityfunctionaltheoryinteractingBrownianparticlesfluctuatinghydrodynamicsmode-couplingdynamicalmultiplicativenoiseelectrolytes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review explains why the Dean-Kawasaki equation matters: it is an exact rewriting of the coupled overdamped Langevin equations for $N$ interacting Brownian particles into a single stochastic equation for the particle density $\rho(x,t)$. No coarse-graining or approximation is needed for the derivation; the noise is multiplicative, with amplitude $\sqrt{2D\rho}$, and the pair interactions enter through a drift term $\mu \nabla \cdot (\rho \int dy \, \rho(y,t) \nabla V(x-y))$. The review then shows how this exact starting point connects to fluctuating hydrodynamics, macroscopic fluctuation theory, mode-coupling theory, and dynamical density functional theory, and how linearizing it reproduces the random phase approximation for the structure factor. A sympathetic reader comes away with the view that SDFT is the common microscopic foundation behind these theories, and that its many physical applications—supercooled liquids, active matter, chemotaxis, electrolytes, tracer diffusion—are variations on solving one equation.

What carries the argument

The central object is the empirical density $\rho(x,t)=\sum_{\alpha=1}^N \delta(x-r_\alpha(t))$, a distribution-valued stochastic field. The key identity is Dean's derivation: applying Itô's lemma to a test function of each particle position and summing over particles converts the coupled Langevin equations into a single stochastic partial differential equation for $\rho$, with Gaussian white noise $\xi(x,t)$ and the square-root density factor $\sqrt{2D\rho}$ becoming the multiplicative noise term. This object carries the entire argument: because it is exact, every later theory—Kawasaki's Fokker-Planck equation for the probability functional, DDFT's mean-field closure, MCT's memory kernels, the linearized fluctuating-hydrodynamics equations—can be viewed as an approximation or projection of the same underlying stochastic density dynamics.

What would settle it

A Brownian dynamics simulation of $N$ non-interacting particles, with the empirical density coarse-grained over a small volume, should show Poissonian equal-time statistics if the DK equation is exactly equivalent to the Langevin dynamics; any measurable deviation from the Poisson prediction for the third or fourth cumulant would refute the claimed exactness.

Watch

Extended reading notes

Core claim

The central claim this review presents and defends is that the macroscopic density of a suspension of $N$ interacting Brownian particles is not an emergent or approximate quantity: the empirical density $\rho(x,t)=\sum_{\alpha=1}^N \delta(x-r_\alpha(t))$ satisfies, without approximation, the stochastic partial differential equation $\partial_t \rho = D \nabla^2 \rho + \nabla \cdot (\xi \sqrt{2D\rho}) + \mu \nabla \cdot (\rho \int dy \, \rho(y,t)\nabla V(x-y))$. This Dean-Kawasaki equation is equivalent to the original $N$-body Langevin dynamics, preserves the particle-entity property, and contains two intrinsic nonlinearities: the pairwise interaction term $\propto \rho^2$ and the multiplicative noise $\propto \sqrt{\rho}$. All later uses of SDFT—linearizing around a uniform state, expanding around a metastable state, or numerically integrating with finite-volume schemes—are efforts to extract predictions from this exact equation, and the review argues that the same equation sits underneath fluctuating hydrodynamics, macroscopic fluctuation theory, mode-coupling theory, and dynamical density functional theory, which differ only in which average, closure, or coarse-graining is applied.

Load-bearing premise

The application program assumes that linearizing the Dean-Kawasaki equation around a uniform or metastable density is valid for dense, interacting, and out-of-equilibrium systems such as electrolytes and active matter, even though the unlinearized equation is rigorously well-posed only for a discrete set of diffusion coefficients.

Editorial extensions

If this is right

Linearizing the DK equation around a uniform density $\rho_0$ yields a Gaussian theory whose static structure factor is the random phase approximation $S(q)=(1+\rho_0 \tilde V(q)/k_B T)^{-1}$, giving a dynamical extension of that classic closure.
Because the DK equation is exact, any theory that starts from a closure of the BBGKY hierarchy—DDFT's adiabatic approximation, MCT's mode-coupling decoupling—can be rederived or compared against the same starting point; MCT has in fact been rederived from the DK equation up to fluctuation-dissipation constraints.
The same linearized equation, when applied to charged species, reproduces Debye-Hückel-Onsager conductivity of dilute electrolytes and yields density correlations between ionic species, so SDFT provides a single framework for electrolyte transport and fluctuations.
Extensions of the DK equation to active particles, hydrodynamic interactions, inertia, chemical reactions, and stochastic resetting mean that the exact-reformulation result is not limited to passive identical overdamped colloids.
Numerically, finite-volume and positivity-preserving schemes for the DK equation give access to density fluctuations and correlation functions that DDFT, being deterministic, cannot produce.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If one treats the Gaussian-kernel regularization $\rho_\epsilon$ as the physically meaningful field, then smoothed-particle hydrodynamics becomes a direct numerical scheme for SDFT; the review reports the regularization results but does not draw this practical conclusion.
A natural test of the linearized theory is to measure the static structure factor of a concentrated electrolyte out of equilibrium and compare with the RPA prediction $S(q)=(1+\rho_0\tilde V(q)/k_B T)^{-1}$; the review lists electrolyte applications but leaves such a direct test to future work.
The discrete-set well-posedness results suggest that the regime where the DK equation is mathematically well-defined may be exactly the regime where its linearization is least needed; that tension is reported in Section IV but not resolved.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Illien's review is exactly what it claims to be: a pedagogical survey of the Dean-Kawasaki equation and stochastic density functional theory. It makes no new scientific claims, and the derivations are standard. What it does well is three things. First, the presentation of Dean's derivation in Section II is clear and correct, including the subtle step where the sum of particle noises is replaced by a √ρ noise with the same covariance. Second, it distinguishes carefully between the different densities: Kawasaki's coarse-grained ρ-hat, Dean's distributional ρ, and DDFT's ensemble-averaged ρ-bar. That distinction is the source of a lot of confusion in the literature, and this review sorts it out explicitly. Third, the review is honest about the mathematical status of the equation. Section IV states plainly that the DK equation is only well-posed for a discrete set of diffusion coefficients, and Section VI B acknowledges that linearization around a uniform state relies on a Fourier transform that is a strong limitation for short-range potentials. That honesty is real and useful.

The soft spots are the ones you'd expect from a single-author review. There are no new results, and the novelty is organizational rather than scientific. The review leans on the author's own recent work in a few places (Refs. [111], [193], [194]), but those citations are used to point to recent results, not to prop up the derivation, so I don't see that as a problem. The weakest step is the one the reader flagged: the linearized equation (31) is used to describe electrolytes, active matter, and tracer problems, and the review does not establish that this linearization is valid for those systems. It does flag the Fourier limitation, but for active matter and nonreciprocal mixtures the validity is simply assumed. That's a soft spot, not a fatal flaw, because the review is a survey and the limitations are partially acknowledged.

Overall, this is a reliable, well-organized review. It deserves to be published after minor revision. The target audience is graduate students and researchers entering SDFT; for them it will be genuinely useful. I would send it to peer review, and I'd cite it as a reference for the derivation and the connections between SDFT, DDFT, and MCT.

Referee Report

0 major / 5 minor

Summary. This manuscript is a comprehensive review of the Dean–Kawasaki (DK) equation and its role as the basis of stochastic density functional theory (SDFT). The author presents Dean's derivation of the stochastic partial differential equation for the empirical density from the overdamped Langevin dynamics, contrasts it with Kawasaki's earlier coarse-grained approach, and clarifies the distinct physical meanings of the densities appearing in the two formulations. The review also situates SDFT relative to fluctuating hydrodynamics, macroscopic fluctuation theory, mode-coupling theory, and dynamical density functional theory; summarizes the mathematical ill-posedness results for the DK equation; describes extensions (inertia, hydrodynamic interactions, active particles, mixtures, reactions, resetting); discusses exact, perturbative, and numerical solution strategies; and surveys applications ranging from supercooled liquids and electrolytes to tracer diffusion and machine learning.

Significance. The review is valuable as a pedagogical and reference resource. It is carefully organized, and its central claim—that the DK equation is an exact (formal) reformulation of the N-particle Langevin dynamics—is properly qualified: the paper explicitly notes that the density is distribution-valued and that rigorous well-posedness holds only for a discrete set of diffusion coefficients. The author is commendably clear about the distinctions among the exact Dean density, the coarse-grained Kawasaki density, and the ensemble-averaged DDFT density, and about the limitations of the linearized DK equation (e.g., the Fourier-transform restriction for short-range potentials). The bibliography is extensive and up to date, including recent mathematical and active-matter literature. These strengths make the review a reliable entry point for researchers entering the field.

minor comments (5)

[I C, II A, II B, VII B] There are several typographical errors that should be corrected, for example 'recenlty' in Section I C, 'particulary' in Section II A, 'Browian' in Section II B, and 'articial' and 'etablish' in Section VII B.
[II B] The statement that the noise term Ξ(x,t) is Gaussian 'because one can average over the noises η_α independently' is too terse, since the weighting factors ρ_α depend on the same noises; the replacement of Ξ by ∇·(√ρ ξ) holds in law after averaging over the noise history, and I recommend reformulating this sentence to avoid a possible misreading.
[IV (Eq. 25)] In Eq. (25), the Gaussian kernel w_ε(y) should be typeset with an explicit exponential and the appropriate normalization for the spatial dimension, for example w_ε(y) = (2π ε²)^{-d/2} exp(-y²/(2ε²)), rather than the current expression with a floating 'e'.
[VI B 2 (Eq. 36)] In Eq. (36), the notation lim_{t→∞} before δF/δρ is confusing for a time-independent metastable state; the equilibrium condition δF/δρ|_{ρ=ρ*} = μ is sufficient.
[II B (item i)] The claim that the derivation of Eq. (13) does not rely on any approximation would be easier to reconcile with the later discussion of ill-posedness (Section IV) if the word 'formal' were used explicitly in that sentence, since a reader could otherwise mistake the distributional equivalence for a classical SPDE equivalence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the DK derivation is self-contained and author self-citations are not load-bearing.

full rationale

The central claim is the formal exactness of the Dean-Kawasaki equation, Eq. (13), with respect to the coupled overdamped Langevin equations, Eq. (3). The derivation chain in Sec. II B is explicit and self-contained: Ito's lemma is applied to Eq. (3), the empirical density is defined in Eq. (12), the equations are summed over particles, and the replacement of the summed particle noise by a single Gaussian noise field is justified by a stated variance-matching lemma. No step defines the DK equation in terms of its own conclusion, and no parameter is fitted to a target quantity and then renamed as a prediction. The paper explicitly acknowledges the main mathematical caveat, namely that the unregularized DK equation is only well-posed for a discrete set of diffusion coefficients (Sec. IV), and it also flags the limitation of the Fourier-space linearization for short-range potentials (Sec. VI B 1). Applications are benchmarked against independent classical results such as the random phase approximation structure factor and Debye-Huckel-Onsager conductivity, rather than against the author's own previous claims. The author's self-citations, for example Refs. [111], [193], and [194], serve as literature pointers to extensions and applications; they do not carry the derivation of Eq. (13) or any uniqueness or existence assertion on which the review's central claim depends. No circular step can be exhibited with a specific reduction of an equation to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

This review does not introduce new parameters or entities. Its derivations rely on standard Ito calculus and on the formal manipulation of singular densities, as well as on approximations such as local equilibrium in Kawasaki's approach. The list below captures these background assumptions.

assumptions (4)

standard math Ito's lemma applied to the density field and test functions
Used in Section II B to derive Eq. (7) and the subsequent equation for the single-particle density.
domain assumption The collective noise term is Gaussian with variance given by the delta-correlated field xi
Invoked in Section II B after Eq. (13); assumes the sum of independent particle noises converges to a Gaussian white noise in distribution.
domain assumption The singular identity rho_alpha(x,t) rho_alpha(x',t) = delta(x-x') rho_alpha(x,t)
Used in Section II B to compute the variance of the noise; treats products of delta functions formally.
domain assumption Local equilibrium hypothesis in Kawasaki's coarse-graining
Stated in Section II A to replace the N-body distribution by an equilibrium form compatible with the coarse-grained density.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Dean-Kawasaki equation and stochastic density functional theory." pith.science (2026). https://pith.science/paper/IPFIAGFB

@misc{pith2026241113467,
  author       = {Pith},
  title        = {Pith review of: The Dean-Kawasaki equation and stochastic density functional theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPFIAGFB}},
  note         = {Machine review of arXiv:2411.13467}
}

read the original abstract

The Dean-Kawasaki (DK) equation, which is at the basis of stochastic density functional theory (SDFT), was proposed in the mid-nineties to describe the evolution of the density of interacting Brownian particles, which can represent a large number of systems such as colloidal suspensions, supercooled liquids, polymer melts, biological molecules, active or chemotactic particles, or ions in solution. This theoretical framework, which can be summarized as a mathematical reformulation of the coupled overdamped Langevin equations that govern the dynamics of the particles, has attracted a significant amount of attention during the past thirty years. In this review, I present the context in which this framework was introduced, and I recall the main assumptions and calculation techniques that are employed to derive the DK equation. Then, in the broader context of statistical mechanics, I show how SDFT is connected to other theories, such fluctuating hydrodynamics, macroscopic fluctuation theory, or mode-coupling theory. The mathematical questions that are raised by the DK equation are presented in a non-specialist language. In the last parts of the review, I show how the original result was extended in several directions, I present the different strategies and approximations that have been employed to solve the DK equation, both analytically and numerically. I finally list the different situations where SDFT was employed to describe the fluctuations of Brownian suspensions, from the physics of active matter to the description of charged particles and electrolytes.

Figures

Figures reproduced from arXiv: 2411.13467 by the authors.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation
math.PR 2024-11 conditional novelty 7.0 of 10

Free massive particle systems, singular-drift Dean-Kawasaki equations, Wasserstein diffusions, and metric-measure Brownian motions are identified as a single process for any ultracontractive reversible diffusion.
Dynamic correlations in a polar fluid: confronting stochastic density functional theory to simulations
cond-mat.soft 2025-07 conditional novelty 6.0 of 10

SDFT with a Kirkwood-factor renormalization quantitatively reproduces longitudinal and transverse polarization dynamics in the Stockmayer fluid, as verified against Brownian dynamics simulations.
Self-diffusion anomalies of an odd tracer in soft-core media
cond-mat.stat-mech 2024-11 conditional novelty 6.0 of 10

For an odd tracer in a Gaussian-core medium, the sign of the interaction correction to self-diffusion is set by the oddness parameter κ, switching from suppression to enhancement at κ=1, which inverts the GCM self-dif...

Reference graph

Works this paper leans on

211 extracted references · 69 canonical work pages · cited by 3 Pith papers

[111]

P. C. Bressloff, A generalized Dean–Kawasaki equa- tion for an interacting Brownian gas in a partially ab- sorbing medium, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480, 20230915 (2024)

2024
[193]

Y. Wang, D. Dean, S. Marbach, and R. Zakine, Inter- actions enhance dispersion in fluctuating channels via emergent flows, Journal of Fluid Mechanics 972, A8 (2023)

work page 2023
[194]

P. L. Muzzeddu, E. Kalz, A. Gambassi, A. Sharma, and R. Metzler, Self-diffusion anomalies of an odd tracer in soft-core media, New Journal of Physics 27, 033025 (2025)

work page 2025
[1]

Writing ρ(x, t) = ρ0 + √ρ0ϕ(x, t), (29) Eq

Linearization around a constant, uniform state A natural choice for ρ∗(x) is the constant, uniform value ρ0 = N/V, where V is the volume of the system [45, 112, 113]. Writing ρ(x, t) = ρ0 + √ρ0ϕ(x, t), (29) Eq. (13) becomes, after having divided both sides by√ρ0: ∂tϕ(x, t) = D∇2ϕ(x, t) + µρ0∇ ·[(ϕ ∗ ∇V )(x, t)] + µ√ρ0∇ ·[ϕ(x, t)(ϕ ∗ ∇V )(x, t)] + √ 2D∇ · ...
[2]

a state which is such that: lim t→∞ δF δρ(x, t) ρ=ρ∗ = µ, (36) where µ is the chemical potential at which the system is maintained

Linearization around a metastable state An alternative to the linearization around a con- stant uniform state, is the linearization around some metastable state of the dynamics ρ∗, i.e. a state which is such that: lim t→∞ δF δρ(x, t) ρ=ρ∗ = µ, (36) where µ is the chemical potential at which the system is maintained. A linear equation satisfied by the pert...
[3]

Brown, XXVII

R. Brown, XXVII. A brief account of microscopical ob- servations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in or- ganic and inorganic bodies, The Philosophical Magazine 4, 161 (1828)
[4]

A. Einstein, ¨Uber die von der molekularkinetischen The- orie der W¨ arme geforderte Bewegung von in ruhen- den Fl¨ ussigkeiten suspendierten Teilchen, Annalen der Physik 17, 549 (1905)

1905
[5]

von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Phys. 21, 759 (1906)

1906

Show all 211 references

[6]

Langevin, Sur la th´ eorie du mouvement brownien, Comptes rendus de l’Acad´ emie des Sciences (Paris)146, 530 (1908)

P. Langevin, Sur la th´ eorie du mouvement brownien, Comptes rendus de l’Acad´ emie des Sciences (Paris)146, 530 (1908)

1908
[7]

C. W. Gardiner, Handbook of Stochastic Methods (Springer, 1985)

1985
[8]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981)

1981
[9]

M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Oxford University Press, 1987)

1987
[10]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer Berlin Hei- delberg, Berlin, Heidelberg, 1992)

1992
[11]

Leimkuhler and C

B. Leimkuhler and C. Matthews, Rational construc- tion of stochastic numerical methods for molecular sam- pling, Applied Mathematics Research eXpress 2013, 34 (2013)

2013
[12]

Rackauckas and Q

C. Rackauckas and Q. Nie, Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory, Discrete & Con- tinuous Dynamical Systems - B 22, 2731 (2017)

2017
[13]

Samm¨ uller and M

F. Samm¨ uller and M. Schmidt, Adaptive Brownian Dy- namics, The Journal of Chemical Physics 155, 134107 (2021)

2021
[14]

A. J. Archer and M. Rauscher, Dynamical density functional theory for interacting Brownian particles: Stochastic or deterministic?, Journal of Physics A: Mathematical and General 37, 9325 (2004)

2004
[15]

M. D. Fabian, B. Shpiro, E. Rabani, D. Neuhauser, and R. Baer, Stochastic density functional theory, WIREs Computational Molecular Science 9, e1412 (2019)

2019
[16]

G¨ otze,Complex Dynamics of Glass-Forming Liquids (Oxford University Press, 2009)

W. G¨ otze,Complex Dynamics of Glass-Forming Liquids (Oxford University Press, 2009)

2009
[17]

Gotze and L

W. Gotze and L. Sjogren, Relaxation processes in su- percooled liquids, Reports on Progress in Physics 55, 241 (1992)

1992
[18]

Kawasaki, Stochastic model of slow dynamics in su- percooled liquids and dense colloidal suspensions, Phys- ica A: Statistical Mechanics and its Applications 208, 35 (1994)

K. Kawasaki, Stochastic model of slow dynamics in su- percooled liquids and dense colloidal suspensions, Phys- ica A: Statistical Mechanics and its Applications 208, 35 (1994)

1994
[19]

Mori, Transport, Collective Motion, and Brownian Motion, Progress of Theoretical Physics 33, 423 (1965)

H. Mori, Transport, Collective Motion, and Brownian Motion, Progress of Theoretical Physics 33, 423 (1965)

1965
[20]

Zwanzig, Memory Effects in Irreversible Thermody- namics, Physical Review 124, 983 (1961)

R. Zwanzig, Memory Effects in Irreversible Thermody- namics, Physical Review 124, 983 (1961)

1961
[21]

Munakata, Liquid Instability and Freezing - Reduc- tive Perturbation Approach, J

T. Munakata, Liquid Instability and Freezing - Reduc- tive Perturbation Approach, J. Phys. Soc. Jpn.43, 1723 (1977)

1977
[22]

Munakata, A Dynamical Extension of Density Func- tional Theory, J

T. Munakata, A Dynamical Extension of Density Func- tional Theory, J. Phys. Soc. Jpn. 58, 2434 (1989)

1989
[23]

Bagchi, Stability of a supercooled liquid to periodic density waves and dynamics of freezing, Physica A: Sta- tistical Mechanics and its Applications 145, 273 (1987)

B. Bagchi, Stability of a supercooled liquid to periodic density waves and dynamics of freezing, Physica A: Sta- tistical Mechanics and its Applications 145, 273 (1987)

1987
[24]

D. S. Dean, Langevin equation for the density of a sys- tem of interacting Langevin processes, J. Phys. A: Math. Gen. 29, L613 (1996)

1996
[25]

Bothe, L

M. Bothe, L. Cocconi, Z. Zhen, and G. Pruessner, Par- ticle entity in the Doi–Peliti and response field for- malisms, Journal of Physics A: Mathematical and The- oretical 56, 175002 (2023)

2023
[26]

Frusawa and R

H. Frusawa and R. Hayakawa, On the controversy over the stochastic density functional equations, Journal of Physics A: Mathematical and General 33, L155 (2000)

2000
[27]

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Course of Theoretical Physics. Vol. 9: Statistical Physics (Part 2) , 2nd ed. (Pergamon Press, Oxford, 1980)

1980
[28]

R. F. Fox, Gaussian stochastic processes in physics, Physics Reports 48, 179 (1978)

1978
[29]

Spohn, Large-Scale Dynamics of Interacting Parti- cles (Springer, 1991)

H. Spohn, Large-Scale Dynamics of Interacting Parti- cles (Springer, 1991)

1991
[30]

Donev, E

A. Donev, E. Vanden-Eijnden, A. Garcia, and J. Bell, On the accuracy of finite-volume schemes for fluctuat- ing hydrodynamics, Communications in Applied Math- ematics and Computational Science 5, 149 (2010)

2010
[31]

Delong, B

S. Delong, B. E. Griffith, E. Vanden-Eijnden, and A. Donev, Temporal integrators for fluctuating hydro- dynamics, Physical Review E 87, 033302 (2013)

2013
[32]

Delong, Y

S. Delong, Y. Sun, B. E. Griffith, E. Vanden-Eijnden, and A. Donev, Multiscale temporal integrators for fluc- tuating hydrodynamics, Physical Review E 90, 063312 (2014)

2014
[33]

J. A. De La Torre, P. Espa˜ nol, and A. Donev, Fi- nite element discretization of non-linear diffusion equa- tions with thermal fluctuations, The Journal of Chemi- cal Physics 142, 094115 (2015)

2015
[34]

C. Kim, A. Nonaka, J. B. Bell, A. L. Garcia, and 16 A. Donev, Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach, The Journal of Chemical Physics 146, 124110 (2017)

2017
[35]

B. Derrida, Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current, Journal of Statistical Mechanics: Theory and Experi- ment 2007, P07023 (2007)

2007
[36]

Bertini, A

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Fluctuations in Stationary Nonequilib- rium States of Irreversible Processes, Physical Review Letters 87, 040601 (2001)

2001
[37]

Bertini, A

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic Fluctuation Theory for Stationary Non-Equilibrium States, J. Stat. Phys. 107, 635 (2002)

2002
[38]

Bertini, A

L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Macroscopic fluctuation theory, Re- views of Modern Physics 87, 593 (2015)

2015
[39]

Derrida and A

B. Derrida and A. Gerschenfeld, Current Fluctuations in One Dimensional Diffusive Systems with a Step Initial Density Profile, Journal of Statistical Physics 137, 978 (2009)

2009
[40]

P. L. Krapivsky, K. Mallick, and T. Sadhu, Large Devi- ations in Single-File Diffusion, Physical Review Letters 113, 078101 (2014)

2014
[41]

P. L. Krapivsky, K. Mallick, and T. Sadhu, Tagged Particle in Single-File Diffusion, Journal of Statistical Physics 160, 885 (2015)

2015
[42]

Poncet, A

A. Poncet, A. Grabsch, P. Illien, and O. B´ enichou, Generalized Correlation Profiles in Single-File Systems, Physical Review Letters 127, 220601 (2021)

2021
[43]

Grabsch, A

A. Grabsch, A. Poncet, P. Rizkallah, P. Illien, and O. B´ enichou, Exact closure and solution for spatial cor- relations in single-file diffusion, Science Advances 8, eabm5043 (2022)

2022
[44]

Mallick, H

K. Mallick, H. Moriya, and T. Sasamoto, Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process, Physical Review Letters 129, 40601 (2022)

2022
[45]

Grabsch and O

A. Grabsch and O. B´ enichou, Tracer Diffusion beyond Gaussian Behavior: Explicit Results for General Single- File Systems, Physical Review Letters 132, 217101 (2024)

2024
[46]

M. A. Dur´ an-Olivencia, P. Yatsyshin, B. D. Goddard, and S. Kalliadasis, General framework for fluctuat- ing dynamic density functional theory, New Journal of Physics 19, 123022 (2017)

2017
[47]

Chavanis, Hamiltonian and Brownian systems with long-range interactions: V

P.-H. Chavanis, Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations, Physica A: Statis- tical Mechanics and its Applications 387, 5716 (2008)

2008
[48]

Velenich, C

A. Velenich, C. Chamon, L. F. Cugliandolo, and D. Kreimer, On the Brownian gas: A field theory with a Poissonian ground state, Journal of Physics A: Math- ematical and Theoretical 41, 235002 (2008)

2008
[49]

H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proceedings of the National Academy of Sciences 56, 1907 (1966)

1966
[50]

Bouchet, K

F. Bouchet, K. Gawedzki, and C. Nardini, Perturbative Calculation of Quasi-Potential in Non-equilibrium Dif- fusions: A Mean-Field Example, Journal of Statistical Physics 163, 1157 (2016)

2016
[51]

Chavanis, The Brownian mean field model, The European Physical Journal B 87, 120 (2014)

P.-H. Chavanis, The Brownian mean field model, The European Physical Journal B 87, 120 (2014)

2014
[52]

J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 2nd ed. (Academic Press, 1986)

1986
[53]

D. R. Reichman and P. Charbonneau, Mode-coupling theory, Journal of Statistical Mechanics: Theory and Experiment 2005, P05013 (2005)

2005
[54]

Szamel, Mode-coupling theory and beyond: A dia- grammatic approach, Progress of Theoretical and Ex- perimental Physics 2013, 012J01 (2013)

G. Szamel, Mode-coupling theory and beyond: A dia- grammatic approach, Progress of Theoretical and Ex- perimental Physics 2013, 012J01 (2013)

2013
[55]

L. M. C. Janssen, Mode-Coupling Theory of the Glass Transition: A Primer, Frontiers in Physics 6, 97 (2018)

2018
[56]

Mayer, K

P. Mayer, K. Miyazaki, and D. R. Reichman, Coopera- tivity beyond Caging: Generalized Mode-Coupling The- ory, Physical Review Letters 97, 095702 (2006)

2006
[57]

G. F. Mazenko, Fundamental theory of statistical par- ticle dynamics, Physical Review E 81, 061102 (2010)

2010
[58]

G. F. Mazenko, Smoluchowski dynamics and the ergodic-nonergodic transition, Physical Review E 83, 041125 (2011)

2011
[59]

S. P. Das and G. F. Mazenko, Newtonian Kinetic The- ory and the Ergodic-Nonergodic Transition, Journal of Statistical Physics 152, 159 (2013)

2013
[60]

P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical Dynamics of Classical Systems, Physical Review A 8, 423 (1973)

1973
[61]

H.-K. Janssen, On a Lagrangean for classical field dy- namics and renormalization group calculations of dy- namical critical properties, Zeitschrift f¨ ur Physik B Con- densed Matter and Quanta 23, 377 (1976)

1976
[62]

De Dominicis and L

C. De Dominicis and L. Peliti, Field-theory renormal- ization and critical dynamics above T c : Helium, anti- ferromagnets, and liquid-gas systems, Physical Review B 18, 353 (1978)

1978
[63]

Miyazaki and D

K. Miyazaki and D. R. Reichman, Mode-coupling theory and the fluctuation–dissipation theorem for nonlinear Langevin equations with multiplicative noise, Journal of Physics A: Mathematical and General 38, L343 (2005)

2005
[64]

T. H. Nishino and H. Hayakawa, Fluctuation- dissipation-relation-preserving field theory of the glass transition in terms of fluctuating hydrodynamics, Phys- ical Review E 78, 061502 (2008)

2008
[65]

S. P. Das, Dynamic transition in a Brownian fluid: Role of fluctuation–dissipation constraints, Journal of Statis- tical Mechanics: Theory and Experiment 2020, 023208 (2020)

2020
[66]

Andreanov, G

A. Andreanov, G. Biroli, and A. Lef` evre, Dynamical field theory for glass-forming liquids, self-consistent re- summations and time-reversal symmetry, Journal of Statistical Mechanics: Theory and Experiment 2006, P07008 (2006)

2006
[67]

B. Kim, K. Kawasaki, H. Jacquin, and F. Van Wijland, Equilibrium dynamics of the Dean-Kawasaki equation: Mode-coupling theory and its extension, Physical Re- view E 89, 012150 (2014)

2014
[68]

Kim and K

B. Kim and K. Kawasaki, A fluctuation-dissipation relationship-preserving field theory for interacting Brownian particles: One-loop theory and mode coupling theory, Journal of Statistical Mechanics: Theory and Experiment 2008, P02004 (2008)

2008
[69]

A. J. Archer, Dynamical density functional theory for dense atomic liquids, Journal of Physics Condensed Matter 18, 5617 (2006)

2006
[70]

te Vrugt, H

M. te Vrugt, H. L¨ owen, and R. Wittkowski, Classical dynamical density functional theory: From fundamen- tals to applications, Adv. Phys. 69, 121 (2020). 17

2020
[71]

U. M. B. Marconi and P. Tarazona, Dynamic density functional theory of fluids, J. Chem. Phys. 110, 8032 (1999)

1999
[72]

D. A. McQuarrie, Statistical Mechanics (Harper and Row, 1976)

1976
[73]

Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Advances in Physics 28, 143 (1979)

R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Advances in Physics 28, 143 (1979)

1979
[74]

Evans, Density functionals in the theory of nonuni- form fluids, in Fundamentals of Inhomogeneous Fluids (Marcel Dekker: New York, 1992) pp

R. Evans, Density functionals in the theory of nonuni- form fluids, in Fundamentals of Inhomogeneous Fluids (Marcel Dekker: New York, 1992) pp. 85–176

1992
[75]

Lovett, C

R. Lovett, C. Y. Mou, and F. P. Buff, The structure of the liquid–vapor interface, The Journal of Chemical Physics 65, 570 (1976)

1976
[76]

Yoshimori, Microscopic derivation of time-dependent density functional methods, Physical Review E 71, 031203 (2005)

A. Yoshimori, Microscopic derivation of time-dependent density functional methods, Physical Review E 71, 031203 (2005)

2005
[77]

Espa˜ nol and H

P. Espa˜ nol and H. L¨ owen, Derivation of dynamical density functional theory using the projection opera- tor technique, The Journal of Chemical Physics 131, 244101 (2009)

2009
[78]

Schmidt and J

M. Schmidt and J. M. Brader, Power functional theory for Brownian dynamics, J. Chem. Phys. 138, 214101 (2013)

2013
[79]

J. F. Lutsko and M. Oettel, Reconsidering power func- tional theory, J. Chem. Phys. 155, 094901 (2021)

2021
[80]

Schmidt, Power functional theory for many-body dy- namics, Reviews of Modern Physics 94, 015007 (2022)

M. Schmidt, Power functional theory for many-body dy- namics, Reviews of Modern Physics 94, 015007 (2022)

2022
[81]

De Las Heras, T

D. De Las Heras, T. Zimmermann, F. Samm¨ uller, S. Hermann, and M. Schmidt, Perspective: How to over- come dynamical density functional theory, Journal of Physics: Condensed Matter 35, 271501 (2023)

2023
[82]

Kawasaki, Interpolation of stochastic and determin- istic reduced dynamics, Physica A: Statistical Mechan- ics and its Applications 362, 249 (2006)

K. Kawasaki, Interpolation of stochastic and determin- istic reduced dynamics, Physica A: Statistical Mechan- ics and its Applications 362, 249 (2006)

2006
[83]

Chavanis, Brownian particles with long- and short-range interactions, Physica A: Statistical Mechan- ics and its Applications 390, 1546 (2011)

P.-H. Chavanis, Brownian particles with long- and short-range interactions, Physica A: Statistical Mechan- ics and its Applications 390, 1546 (2011)

2011
[84]

Chavanis, The Generalized Stochastic Smolu- chowski Equation, Entropy

P.-H. Chavanis, The Generalized Stochastic Smolu- chowski Equation, Entropy. An International and Inter- disciplinary Journal of Entropy and Information Studies 21, 1006 (2019)

2019
[85]

Konarovskyi, T

V. Konarovskyi, T. Lehmann, and M. K. von Renesse, Dean-kawasaki dynamics: Ill-posedness vs. triviality, Electron. Commun. Probab. 24, 1 (2019)

2019
[86]

Konarovskyi, T

V. Konarovskyi, T. Lehmann, and M. Von Renesse, On Dean–Kawasaki Dynamics with Smooth Drift Potential, Journal of Statistical Physics 178, 666 (2020)

2020
[87]

Konarovskyi and F

V. Konarovskyi and F. M¨ uller, Dean–Kawasaki equa- tion with initial condition in the space of positive distri- butions, Journal of Evolution Equations 24, 92 (2024)

2024
[88]

L. Wang, Z. Wu, and R. Zhang, Dean-Kawasaki equa- tion with singular interactions and applications to dy- namical Ising-Kac model (2024), arXiv:2207.12774

2024 arXiv
[89]

M¨ uller, M

F. M¨ uller, M. von Renesse, and J. Zimmer, Well- Posedness for Dean-Kawasaki Models of Vlasov-Fokker- Planck Type (2025), arXiv:2411.14334 [math]

2025 arXiv
[90]

Violeau, Fluid Mechanics and the SPH Method: The- ory and Applications (Oxford University Press, 2012)

D. Violeau, Fluid Mechanics and the SPH Method: The- ory and Applications (Oxford University Press, 2012)

2012
[91]

Cornalba, T

F. Cornalba, T. Shardlow, and J. Zimmer, A Regu- larized Dean–Kawasaki Model: Derivation and Anal- ysis, SIAM Journal on Mathematical Analysis 51, 1137 (2019)

2019
[92]

Cornalba, T

F. Cornalba, T. Shardlow, and J. Zimmer, From weakly interacting particles to a regularised Dean–Kawasaki model, Nonlinearity 33, 864 (2020)

2020
[93]

Cornalba, T

F. Cornalba, T. Shardlow, and J. Zimmer, Well- posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions, Journal of Differential Equations 284, 253 (2021)

2021
[94]

Cornalba, J

F. Cornalba, J. Fischer, J. Ingmanns, and C. Raithel, Density fluctuations in weakly interacting particle systems via the Dean-Kawasaki equation (2023), arXiv:2303.00429

2023 arXiv
[95]

Nakamura and A

T. Nakamura and A. Yoshimori, Derivation of the non- linear fluctuating hydrodynamic equation from the un- derdamped Langevin equation, Journal of Physics A: Mathematical and Theoretical 42, 065001 (2009)

2009
[96]

S. P. Das and A. Yoshimori, Coarse-grained forms for equations describing the microscopic motion of particles in a fluid, Physical Review E 88, 043008 (2013)

2013
[97]

L´ opez and A

C. L´ opez and A. Puglisi, Continuum description of finite-size particles advected by external flows: The ef- fect of collisions, Physical Review E 69, 046306 (2004)

2004
[98]

Donev and E

A. Donev and E. Vanden-Eijnden, Dynamic density functional theory with hydrodynamic interactions and fluctuations, The Journal of Chemical Physics 140, 234115 (2014)

2014
[99]

R. P. Pel´ aez, F. B. Usabiaga, S. Panzuela, Q. Xiao, R. Delgado-Buscalioni, and A. Donev, Hydrodynamic fluctuations in quasi-two dimensional diffusion, Journal of Statistical Mechanics: Theory and Experiment 2018, 063207 (2018)

2018
[100]

D. L. Ermak and J. A. McCammon, Brownian dynamics with hydrodynamic interactions, The Journal of Chem- ical Physics 69, 1352 (1978)

1978
[101]

L. F. Cugliandolo, P.-M. D´ ejardin, G. S. Lozano, and F. Van Wijland, Stochastic dynamics of collective modes for Brownian dipoles, Physical Review E 91, 032139 (2015)

2015
[102]

Illien, A

P. Illien, A. Carof, and B. Rotenberg, Stochastic den- sity functional theory for ions in a polar solvent (2024), arXiv:2407.17232

2024 arXiv
[103]

D´ emery and D

V. D´ emery and D. S. Dean, The conductivity of strong electrolytes from stochastic density functional theory, J. Stat. Mech. , 023106 (2016)

2016
[104]

Poncet, O

A. Poncet, O. B´ enichou, V. D´ emery, and G. Oshanin, Universal long ranged correlations in driven binary mix- tures, Phys. Rev. Lett. 118, 118002 (2017)

2017
[105]

Jardat, V

M. Jardat, V. Dahirel, and P. Illien, Diffusion of a tracer in a dense mixture of soft particles connected to different thermostats, Phys. Rev. E 106, 064608 (2022)

2022
[106]

Benois, M

A. Benois, M. Jardat, V. Dahirel, V. D´ emery, J. Agudo- Canalejo, R. Golestanian, and P. Illien, Enhanced diffu- sion of tracer particles in nonreciprocal mixtures, Phys- ical Review E 108, 054606 (2023)

2023
[107]

Fodor, H

´E. Fodor, H. Hayakawa, J. Tailleur, and F. Van Wijland, Non-Gaussian noise without memory in active matter, Physical Review E 98, 062610 (2018)

2018
[108]

R. E. Spinney and R. G. Morris, A Dean-Kawasaki equation for reaction diffusion systems driven by Pois- son noise (2024), arXiv:2404.02487

2024 arXiv
[109]

P. C. Bressloff, Global density equations for a popula- tion of actively switching particles, Journal of Physics A: Mathematical and Theoretical 57, 085001 (2024)

2024
[110]

P. C. Bressloff, Global density equations for interact- ing particle systems with stochastic resetting: From 18 overdamped Brownian motion to phase synchronization, Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence 34, 043101 (2024)

2024
[112]

B. Kim, M. Fuchs, and V. Krakoviack, Dynamics of a noninteracting colloidal fluid in a quenched Gaus- sian random potential: A time-reversal-symmetry- preserving field-theoretic approach, J. Stat. Mech. 2020, 023301 (2020)

2020
[113]

Illien and A

P. Illien and A. Carof, Non-Gaussian density fluc- tuations in the Dean-Kawasaki equation (2025), arXiv:2501.16206

2025 arXiv
[114]

D. S. Dean and R. Podgornik, Relaxation of the ther- mal Casimir force between net neutral plates contain- ing Brownian charges, Physical Review E 89, 032117 (2014)

2014
[115]

D´ emery, O

V. D´ emery, O. B´ enichou, and H. Jacquin, Generalized Langevin equations for a driven tracer in dense soft col- loids: Construction and applications, New Journal of Physics 16, 053032 (2014)

2014
[116]

H. C. Andersen and D. Chandler, Mode Expansion in Equilibrium Statistical Mechanics. I. General Theory and Application to the Classical Electron Gas, The Journal of Chemical Physics 53, 547 (1970)

1970
[117]

J. C. Wheeler and D. Chandler, Catastrophe in the Random-Phase Approximation: Critique of a Theory of Phase Transitions, The Journal of Chemical Physics 55, 1645 (1971)

1971
[118]

C. N. Likos, A. Lang, M. Watzlawek, and H. L¨ owen, Cri- terion for determining clustering versus reentrant melt- ing behavior for bounded interaction potentials, Physi- cal Review E 63, 031206 (2001)

2001
[119]

A. Lang, C. N. Likos, M. Watzlawek, and H. L¨ owen, Fluid and solid phases of the Gaussian core model, Jour- nal of Physics: Condensed Matter 12, 5087 (2000)

2000
[120]

A. A. Louis, P. G. Bolhuis, and J. P. Hansen, Mean- field fluid behavior of the Gaussian core model, Physical Review E 62, 7961 (2000)

2000
[121]

Kr¨ uger and D

M. Kr¨ uger and D. S. Dean, A Gaussian theory for fluc- tuations in simple liquids, Journal of Chemical Physics 146, 134507 (2017)

2017
[122]

Kr¨ uger, A

M. Kr¨ uger, A. Solon, V. D´ emery, C. M. Rohwer, and D. S. Dean, Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory, The Journal of Chemical Physics 148, 084503 (2018)

2018
[123]

Jin and D

J. Jin and D. R. Reichman, Perturbative Expansion in Reciprocal Space: Bridging Microscopic and Mesoscopic Descriptions of Molecular Interactions, The Journal of Physical Chemistry B 128, 1061 (2024)

2024
[124]

H. Frusawa, Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean– Kawasaki equation, Journal of Physics A: Mathematical and Theoretical 52, 065003 (2019)

2019
[125]

H. Frusawa, Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: Stochastic density functional theory at strong coupling, Soft Matter 17, 8810 (2021)

2021
[126]

Adhikari, K

R. Adhikari, K. Stratford, M. E. Cates, and A. J. Wag- ner, Fluctuating lattice Boltzmann, Europhysics Letters (EPL) 71, 473 (2005)

2005
[127]

Russo, S

A. Russo, S. P. Perez, M. A. Dur´ an-Olivencia, P. Yat- syshin, J. A. Carrillo, and S. Kalliadasis, A finite-volume method for fluctuating dynamical density functional theory, Journal of Computational Physics 428, 109796 (2021)

2021
[128]

Mendes, A

J. Mendes, A. Russo, S. P. Perez, and S. Kalliadasis, A finite-volume scheme for gradient-flow equations with non-homogeneous diffusion, Computers & Mathematics with Applications 89, 150 (2021)

2021
[129]

Mart ´ ınez-Lera and M

P. Mart ´ ınez-Lera and M. De Corato, A finite element method for stochastic diffusion equations using fluctuat- ing hydrodynamics, Journal of Computational Physics 510, 113098 (2024)

2024
[130]

Magaletti, M

F. Magaletti, M. Gallo, S. P. Perez, J. A. Carrillo, and S. Kalliadasis, A positivity-preserving scheme for fluctuating hydrodynamics, Journal of Computational Physics 463, 111248 (2022)

2022
[131]

Cornalba and J

F. Cornalba and J. Fischer, The Dean–Kawasaki Equa- tion and the Structure of Density Fluctuations in Sys- tems of Diffusing Particles, Archive for Rational Me- chanics and Analysis 247, 76 (2023)

2023
[132]

Cornalba and T

F. Cornalba and T. Shardlow, The regularised iner- tial Dean–Kawasaki equation: Discontinuous Galerkin approximation and modelling for low-density regime, ESAIM: Mathematical Modelling and Numerical Anal- ysis 57, 3061 (2023)

2023
[133]

B. S. Gupta, S. P. Das, and J.-L. Barrat, Time- dependent correlations in a supercooled liquid from non- linear fluctuating hydrodynamics, Physical Review E 83, 041506 (2011)

2011
[134]

Bidhoodi and S

N. Bidhoodi and S. P. Das, Ergodicity and slow diffusion in a supercooled liquid, Physica A: Statistical Mechanics and its Applications 449, 357 (2016)

2016
[135]

Ramaswamy, The Mechanics and Statistics of Active Matter, Annual Review of Condensed Matter Physics 1, 323 (2010)

S. Ramaswamy, The Mechanics and Statistics of Active Matter, Annual Review of Condensed Matter Physics 1, 323 (2010)

2010
[136]

M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrody- namics of soft active matter, Reviews of Modern Physics 85, 1143 (2013)

2013
[137]

Tailleur and M

J. Tailleur and M. Cates, Statistical Mechanics of Inter- acting Run-and-Tumble Bacteria, Physical Review Let- ters 100, 218103 (2008)

2008
[138]

´O Laighl´ eis, M

E. ´O Laighl´ eis, M. R. Evans, and R. A. Blythe, Minimal stochastic field equations for one-dimensional flocking, Physical Review E 98, 062127 (2018)

2018
[139]

Bertin, H

E. Bertin, H. Chat´ e, F. Ginelli, S. Mishra, A. Peshkov, and S. Ramaswamy, Mesoscopic theory for fluctuating active nematics, New Journal of Physics 15, 085032 (2013)

2013
[140]

A. P. Solon, M. E. Cates, and J. Tailleur, Active brown- ian particles and run-and-tumble particles: A compara- tive study, The European Physical Journal Special Top- ics 224, 1231 (2015)

2015
[141]

Martin, J

D. Martin, J. O’Byrne, M. E. Cates, ´E. Fodor, C. Nar- dini, J. Tailleur, and F. Van Wijland, Statistical me- chanics of active Ornstein-Uhlenbeck particles, Physical Review E 103, 032607 (2021)

2021
[142]

Dinelli, J

A. Dinelli, J. O’Byrne, and J. Tailleur, Fluctuating hy- drodynamics of active particles interacting via chemo- taxis and quorum sensing: static and dynamics, J. Phys. A: Math. Theor. 57, 395002 (2024)

2024
[143]

Poncet, O

A. Poncet, O. B´ enichou, V. D´ emery, and D. Nishiguchi, 19 Pair correlation of dilute active Brownian particles: From low-activity dipolar correction to high-activity al- gebraic depletion wings, Physical Review E 103, 012605 (2021)

2021
[144]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49, 435 (1977)

1977
[145]

P. M. Chaikin and T. C. Lubensky, Principles of Con- densed Matter Physics (Cambridge University Press)
[146]

Tjhung, C

E. Tjhung, C. Nardini, and M. E. Cates, Cluster Phases and Bubbly Phase Separation in Active Fluids: Reversal of the Ostwald Process, Phys. Rev. X 8, 31080 (2018)

2018
[147]

GrandPre, K

T. GrandPre, K. Klymko, K. K. Mandadapu, and D. T. Limmer, Entropy production fluctuations encode collec- tive behavior in active matter, Physical Review E 103, 012613 (2021)

2021
[148]

Tociu, ´E

L. Tociu, ´E. Fodor, T. Nemoto, and S. Vaikuntanathan, How Dissipation Constrains Fluctuations in Nonequi- librium Liquids: Diffusion, Structure, and Biased Inter- actions, Physical Review X 9, 41026 (2019)

2019
[149]

Fodor, T

´E. Fodor, T. Nemoto, and S. Vaikuntanathan, Dissipa- tion controls transport and phase transitions in active fluids: Mobility, diffusion and biased ensembles, New Journal of Physics 22, 013052 (2020)

2020
[150]

Rassolov, L

G. Rassolov, L. Tociu, ´E. Fodor, and S. Vaikuntanathan, From predicting to learning dissipation from pair cor- relations of active liquids, The Journal of Chemical Physics 157, 054901 (2022)

2022
[151]

Tociu, G

L. Tociu, G. Rassolov, ´E. Fodor, and S. Vaikuntanathan, Mean-field theory for the structure of strongly interact- ing active liquids, The Journal of Chemical Physics157, 014902 (2022)

2022
[152]

Kuroda and K

Y. Kuroda and K. Miyazaki, Microscopic theory for hyperuniformity in two-dimensional chiral active fluid, Journal of Statistical Mechanics: Theory and Experi- ment 2023, 103203 (2023)

2023
[153]

Damman, V

P. Damman, V. D´ emery, G. Palumbo, and Q. Thomas, Algebraic depletion interactions in two-temperature mixtures (2024), arXiv:2406.11616

2024 arXiv
[154]

A. Y. Grosberg and J.-F. Joanny, Nonequilibrium statis- tical mechanics of mixtures of particles in contact with different thermostats, Physical Review E 92, 032118 (2015)

2015
[155]

Ghimenti, L

F. Ghimenti, L. Berthier, G. Szamel, and F. van Wij- land, Irreversible Boltzmann samplers in dense liquids: Weak-coupling approximation and mode-coupling the- ory, Phys. Rev. E 110, 034604 (2024)

2024
[156]

Dinelli, J

A. Dinelli, J. O’Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Non-reciprocity across scales in active mixtures, Nature Communications 14, 7035 (2023)

2023
[157]

Chavanis, A stochastic Keller–Segel model of chemotaxis, Communications in Nonlinear Science and Numerical Simulation 15, 60 (2010)

P.-H. Chavanis, A stochastic Keller–Segel model of chemotaxis, Communications in Nonlinear Science and Numerical Simulation 15, 60 (2010)

2010
[158]

P. H. Chavanis and L. Delfini, Random transitions de- scribed by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model, Physical Re- view E 89, 032139 (2014)

2014
[159]

Forster, D

D. Forster, D. R. Nelson, and J. Stephen, Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A 16, 732 (1977)

1977
[160]

Medina, T

E. Medina, T. Hwa, M. Kardar, and Y. C. Zhang, Burg- ers equation with correlated noise: Renormalization- group analysis and applications to directed polymers and interface growth, Physical Review A 39, 3053 (1989)

1989
[161]

Gelimson and R

A. Gelimson and R. Golestanian, Collective Dynamics of Dividing Chemotactic Cells, Physical Review Letters 114, 028101 (2015)

2015
[162]

Mahdisoltani, R

S. Mahdisoltani, R. B. A. Zinati, C. Duclut, A. Gam- bassi, and R. Golestanian, Nonequilibrium polarity- induced chemotaxis: Emergent Galilean symmetry and exact scaling exponents, Physical Review Research 3, 013100 (2021)

2021
[163]

Ben Al ` ı Zinati, C

R. Ben Al ` ı Zinati, C. Duclut, S. Mahdisoltani, A. Gam- bassi, and R. Golestanian, Stochastic dynamics of chemotactic colonies with logistic growth, Europhysics Letters 136, 50003 (2021)

2021
[164]

Mahdisoltani and R

S. Mahdisoltani and R. Golestanian, Nonequilibrium phenomena in driven and active Coulomb field theories, Physica A: Statistical Mechanics and its Applications 631, 127947 (2023)

2023
[165]

Van Der Kolk, F

J. Van Der Kolk, F. Raßhofer, R. Swiderski, A. Haldar, A. Basu, and E. Frey, Anomalous Collective Dynamics of Autochemotactic Populations, Physical Review Let- ters 131, 088201 (2023)

2023
[166]

H. S. Samanta and D. Thirumalai, Origin of superdif- fusive behavior in a class of nonequilibrium systems, Physical Review E 99, 032401 (2019)

2019
[167]

H. S. Samanta, Interstitial flows regulate collective cell migration heterogeneity through adhesion, Physical Re- view Research 2, 013048 (2020)

2020
[168]

Kjellander, Statistical Mechanics of Liquids and So- lutions: Intermolecular Forces, Structure and Surface Interactions, 1st ed

R. Kjellander, Statistical Mechanics of Liquids and So- lutions: Intermolecular Forces, Structure and Surface Interactions, 1st ed. (CRC Press, 2019)

2019
[169]

Bonneau, V

H. Bonneau, V. D´ emery, and´E. Rapha¨ el, Temporal re- sponse of the conductivity of electrolytes, J. Stat. Mech. 2023, 073205 (2023)

2023
[170]

Bonneau, V

H. Bonneau, V. D´ emery, and E. Rapha¨ el, Stationary and transient correlations in driven electrolytes, J. Stat. Mech. 2025, 033201 (2025)

2025
[171]

Berthoumieux, V

H. Berthoumieux, V. D´ emery, and A. C. Maggs, Non- monotonic conductivity of aqueous electrolytes: Beyond the first Wien effect (2024)

2024
[172]

Frusawa, Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View, Entropy 22, 34 (2019)

H. Frusawa, Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View, Entropy 22, 34 (2019)

2019
[173]

Hoang Ngoc, B

M.-T. Hoang Ngoc, B. Rotenberg, and S. Marbach, Ionic fluctuations in finite volumes: Fractional noise and hyperuniformity, Faraday Discussions 246, 225 (2023)

2023
[174]

R. Okamoto, Fluctuating hydrodynamics of dilute elec- trolyte solutions: Systematic perturbation calculation of effective transport coefficients governing large-scale dynamics, Journal of Statistical Mechanics: Theory and Experiment 2022, 093203 (2022)

2022
[175]

Poitevin, M

F. Poitevin, M. Delarue, and H. Orland, Beyond Poisson–Boltzmann: Numerical Sampling of Charge Density Fluctuations, The Journal of Physical Chem- istry B 120, 6270 (2016)

2016
[176]

Y. Avni, R. M. Adar, D. Andelman, and H. Orland, Conductivity of Concentrated Electrolytes, Physical Re- view Letters 128, 098002 (2022)

2022
[177]

Y. Avni, D. Andelman, and H. Orland, Conductance of concentrated electrolytes: Multivalency and the Wien effect, The Journal of Chemical Physics 157, 154502 (2022). 20

2022
[178]

Robin, Correlation-induced viscous dissipation in concentrated electrolytes, The Journal of Chemical Physics 160, 064503 (2024)

P. Robin, Correlation-induced viscous dissipation in concentrated electrolytes, The Journal of Chemical Physics 160, 064503 (2024)

2024
[179]

Bernard, M

O. Bernard, M. Jardat, B. Rotenberg, and P. Illien, On analytical theories for conductivity and self-diffusion in concentrated electrolytes, The Journal of Chemical Physics 159, 164105 (2023)

2023
[180]

H. Frusawa, Electric-field-induced oscillations in ionic fluids: A unified formulation of modified Poisson– Nernst–Planck models and its relevance to correlation function analysis, Soft Matter 18, 4280 (2022)

2022
[181]

Kardar and R

M. Kardar and R. Golestanian, The ”friction” of vac- uum, and other fluctuation-induced forces, Rev. Mod. Phys. 71, 1233 (1999)

1999
[182]

B.-S. Lu, D. S. Dean, and R. Podgornik, Out-of- equilibrium thermal Casimir effect between Brown- ian conducting plates, EPL (Europhysics Letters) 112, 20001 (2015)

2015
[183]

D. S. Dean, B.-S. Lu, A. C. Maggs, and R. Podgornik, Nonequilibrium Tuning of the Thermal Casimir Effect, Physical Review Letters 116, 240602 (2016)

2016
[184]

Mahdisoltani and R

S. Mahdisoltani and R. Golestanian, Long-Range Fluctuation-Induced Forces in Driven Electrolytes, Physical Review Letters 126, 158002 (2021)

2021
[185]

Mahdisoltani and R

S. Mahdisoltani and R. Golestanian, Transient fluctuation-induced forces in driven electrolytes after an electric field quench, New Journal of Physics 23, 073034 (2021)

2021
[186]

G. Du, D. S. Dean, B. Miao, and R. Podgornik, Correlation decoupling of Casimir interaction in an electrolyte driven by external electric fields (2024), arXiv:2404.06028

2024 arXiv
[187]

G. Du, D. S. Dean, B. Miao, and R. Podgornik, Re- pulsive thermal van der Waals interaction in multi- species asymmetric electrolytes driven by external elec- tric fields, Physical Review E 111, 044108 (2025)

2025
[188]

D´ emery and D

V. D´ emery and D. S. Dean, Perturbative path-integral study of active- and passive-tracer diffusion in fluctuat- ing fields, Physical Review E 84, 011148 (2011)

2011
[189]

D´ emery, Mean-field microrheology of a very soft colloidal suspension: Inertia induces shear thickening, Physical Review E 91, 062301 (2015)

V. D´ emery, Mean-field microrheology of a very soft colloidal suspension: Inertia induces shear thickening, Physical Review E 91, 062301 (2015)

2015
[190]

D´ emery and´E

V. D´ emery and´E. Fodor, Driven probe under harmonic confinement in a colloidal bath, Journal of Statisti- cal Mechanics: Theory and Experiment 2019, 033202 (2019)

2019
[191]

Martin, C

D. Martin, C. Nardini, M. E. Cates, and ´E. Fodor, Ex- tracting maximum power from active colloidal heat en- gines, EPL (Europhysics Letters) 121, 60005 (2018)

2018
[192]

Feng and Z

M. Feng and Z. Hou, Unraveling on kinesin acceleration in intracellular environments: A theory for active bath, Physical Review Research 5, 013206 (2023)

2023
[195]

Venturelli, P

D. Venturelli, P. Illien, A. Grabsch, and O. B´ enichou, Universal scale-free decay of tracer-bath correlations in $d$-dimensional interacting particle systems (2025), arXiv:2411.09326

2025
[196]

Venturelli, P

D. Venturelli, P. Illien, A. Grabsch, and O. B´ enichou, Dynamics of soft interacting particles on a comb (2025), arXiv:2502.16951

2025 arXiv
[197]

Ooshida, S

T. Ooshida, S. Goto, T. Matsumoto, A. Nakahara, and M. Otsuki, Continuum Theory of Single-File Diffusion in Terms of Label Variable, Journal of the Physical So- ciety of Japan 80, 074007 (2011)

2011
[198]

Ooshida, S

T. Ooshida, S. Goto, T. Matsumoto, A. Nakahara, and M. Otsuki, Analytical calculation of four-point correla- tions for a simple model of cages involving numerous particles, Physical Review E 88, 062108 (2013)

2013
[199]

Ooshida, S

T. Ooshida, S. Goto, T. Matsumoto, and M. Ot- suki, Displacement correlation as an indicator of collective motion in one-dimensional and quasi-one- dimensional systems of repulsive Brownian particles, Modern Physics Letters B 29, 1550221 (2015)

2015
[200]

Ooshida and M

T. Ooshida and M. Otsuki, Two-tag correlations and nonequilibrium fluctuation–response relation in ageing single-file diffusion, Journal of Physics: Condensed Mat- ter 30, 374001 (2018)

2018
[201]

S. M. Abel, Y.-L. Steve Tse, and H. C. Andersenb, Ki- netic theories of dynamics and persistent caging in a one-dimensional lattice gas, Proceedings of the National Academy of Sciences 106, 15142 (2009)

2009
[202]

Ooshida, S

T. Ooshida, S. Goto, T. Matsumoto, and M. Otsuki, Calculation of displacement correlation tensor indicat- ing vortical cooperative motion in two-dimensional col- loidal liquids, Physical Review E 94, 022125 (2016)

2016
[203]

Ooshida, S

T. Ooshida, S. Goto, T. Matsumoto, and M. Otsuki, Insights from Single-File Diffusion into Cooperativity in Higher Dimensions, Biophysical Reviews and Letters 11, 9 (2016)

2016
[204]

Touzo, P

L. Touzo, P. Le Doussal, and G. Schehr, Interacting, running and tumbling: The active Dyson Brownian mo- tion, Europhysics Letters 142, 61004 (2023)

2023
[205]

Le Doussal, Ranked diffusion, delta Bose gas, and Burgers equation, Physical Review E 105, L012103 (2022)

P. Le Doussal, Ranked diffusion, delta Bose gas, and Burgers equation, Physical Review E 105, L012103 (2022)

2022
[206]

Flack, P

A. Flack, P. Le Doussal, S. N. Majumdar, and G. Schehr, Out-of-equilibrium dynamics of repulsive ranked diffusions: The expanding crystal, Physical Re- view E 107, 064105 (2023)

2023
[207]

Dandekar, P

R. Dandekar, P. L. Krapivsky, and K. Mallick, Dynam- ical fluctuations in the Riesz gas, Physical Review E 107, 044129 (2023)

2023
[208]

Dandekar, P

R. Dandekar, P. L. Krapivsky, and K. Mallick, Current fluctuations in the dyson gas (2024), arXiv:2409.06881

2024 arXiv
[209]

Rotskoff and E

G. Rotskoff and E. Vanden-Eijnden, Trainability and Accuracy of Artificial Neural Networks: An Interacting Particle System Approach, Communications on Pure and Applied Mathematics 75, 1889 (2022)

2022
[210]

J. F. Lutsko, A dynamical theory of nucleation for colloids and macromolecules, The Journal of Chemical Physics 136, 034509 (2012)

2012
[211]

J. Liu, J. E. Sprittles, and T. Grafke, Mean first pas- sage times and eyring-kramers formula for fluctuating hydrodynamics (2024), arXiv:2405.13490

2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [111]

P. C. Bressloff, A generalized Dean–Kawasaki equa- tion for an interacting Brownian gas in a partially ab- sorbing medium, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480, 20230915 (2024)

2024

[2] [193]

Y. Wang, D. Dean, S. Marbach, and R. Zakine, Inter- actions enhance dispersion in fluctuating channels via emergent flows, Journal of Fluid Mechanics 972, A8 (2023)

work page 2023

[3] [194]

P. L. Muzzeddu, E. Kalz, A. Gambassi, A. Sharma, and R. Metzler, Self-diffusion anomalies of an odd tracer in soft-core media, New Journal of Physics 27, 033025 (2025)

work page 2025

[4] [1]

Writing ρ(x, t) = ρ0 + √ρ0ϕ(x, t), (29) Eq

Linearization around a constant, uniform state A natural choice for ρ∗(x) is the constant, uniform value ρ0 = N/V, where V is the volume of the system [45, 112, 113]. Writing ρ(x, t) = ρ0 + √ρ0ϕ(x, t), (29) Eq. (13) becomes, after having divided both sides by√ρ0: ∂tϕ(x, t) = D∇2ϕ(x, t) + µρ0∇ ·[(ϕ ∗ ∇V )(x, t)] + µ√ρ0∇ ·[ϕ(x, t)(ϕ ∗ ∇V )(x, t)] + √ 2D∇ · ...

[5] [2]

a state which is such that: lim t→∞ δF δρ(x, t) ρ=ρ∗ = µ, (36) where µ is the chemical potential at which the system is maintained

Linearization around a metastable state An alternative to the linearization around a con- stant uniform state, is the linearization around some metastable state of the dynamics ρ∗, i.e. a state which is such that: lim t→∞ δF δρ(x, t) ρ=ρ∗ = µ, (36) where µ is the chemical potential at which the system is maintained. A linear equation satisfied by the pert...

[6] [3]

Brown, XXVII

R. Brown, XXVII. A brief account of microscopical ob- servations made in the months of June, July and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in or- ganic and inorganic bodies, The Philosophical Magazine 4, 161 (1828)

[7] [4]

A. Einstein, ¨Uber die von der molekularkinetischen The- orie der W¨ arme geforderte Bewegung von in ruhen- den Fl¨ ussigkeiten suspendierten Teilchen, Annalen der Physik 17, 549 (1905)

1905

[8] [5]

von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann

M. von Smoluchowski, Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Phys. 21, 759 (1906)

1906