{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T18:01:07Z","timestamp":1770487267771,"version":"3.49.0"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,25]],"date-time":"2019-12-25T00:00:00Z","timestamp":1577232000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We consider the Dean\u2013Kawasaki (DK) equation of overdamped Brownian particles that forms the basis of the stochastic density functional theory. Recently, the linearized DK equation has successfully reproduced the full Onsager theory of symmetric electrolyte conductivity. In this paper, the linear DK equation is applied to investigate density fluctuations around the ground state distribution of strongly coupled counterions near a charged plate, focusing especially on the transverse dynamics along the plate surface. Consequently, we find a crossover scale above which the transverse density dynamics appears frozen and below which diffusive behavior of counterions can be observed on the charged plate. The linear DK equation provides a characteristic length of the dynamical crossover that is similar to the Wigner\u2013Seitz radius used in equilibrium theory for the 2D one-component plasma, which is our main result. Incidentally, general representations of longitudinal dynamics vertical to the plate further suggest the existence of advective and electrical reverse-flows; these effects remain to be quantitatively investigated.<\/jats:p>","DOI":"10.3390\/e22010034","type":"journal-article","created":{"date-parts":[[2019,12,25]],"date-time":"2019-12-25T11:07:48Z","timestamp":1577272068000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Transverse Density Fluctuations around the Ground State Distribution of Counterions near One Charged Plate: Stochastic Density Functional View"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5567-4315","authenticated-orcid":false,"given":"Hiroshi","family":"Frusawa","sequence":"first","affiliation":[{"name":"Laboratory of Statistical Physics, Kochi University of Technology, Tosa-Yamada, Kochi 782-8502, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,25]]},"reference":[{"key":"ref_1","unstructured":"Israelachvili, J. (2015). Intermolecular and Surface Forces, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1577","DOI":"10.1088\/0034-4885\/65\/11\/201","article-title":"Electrostatic correlations: From plasma to biology","volume":"65","author":"Levin","year":"2002","journal-title":"Rep. Prog. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(80)90022-8","article-title":"Statistical mechanics of simple Coulomb systems","volume":"59","author":"Baus","year":"1980","journal-title":"Phys. Rep."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1103\/RevModPhys.81.1353","article-title":"Complex plasmas: An interdisciplinary research field","volume":"81","author":"Morfill","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1002\/ctpp.201500104","article-title":"Internal energy of the elassical two-and three-dimensional one-component-plasma","volume":"56","author":"Khrapak","year":"2016","journal-title":"Contrib. Plasm. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s101890170039","article-title":"Electrostatistics of counter-ions at and between planar charged walls: From Poisson-Boltzmann to the strong-coupling theory","volume":"5","author":"Netz","year":"2001","journal-title":"Eur. Phys. J. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1103\/RevModPhys.74.329","article-title":"Colloquium: The physics of charge inversion in chemical and biological systems","volume":"74","author":"Grosberg","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"150901","DOI":"10.1063\/1.4824681","article-title":"Perspective: Coulomb fluids-weak coupling, strong coupling, in between and beyond","volume":"139","author":"Naji","year":"2013","journal-title":"J. Chem. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1140\/epje\/i2001-10091-9","article-title":"Simulations of counterions at charged plates","volume":"8","author":"Moreira","year":"2002","journal-title":"Eur. Phys. J. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3134","DOI":"10.1080\/00268976.2018.1471234","article-title":"A correlation-hole approach to the electric double layer with counter-ions only","volume":"116","author":"Palaia","year":"2018","journal-title":"Mol. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4040","DOI":"10.1039\/C8SM00571K","article-title":"Strong-coupling theory of counterions between symmetrically charged walls: From crystal to fluid phases","volume":"14","author":"Trulsson","year":"2018","journal-title":"Soft Matter"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8032","DOI":"10.1063\/1.478705","article-title":"Dynamic density functional theory of fluids","volume":"110","author":"Marconi","year":"1999","journal-title":"J. Chem. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"L613","DOI":"10.1088\/0305-4470\/29\/24\/001","article-title":"Langevin equation for the density of a system of interacting Langevin processes","volume":"29","author":"Dean","year":"1996","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"L155","DOI":"10.1088\/0305-4470\/33\/15\/101","article-title":"On the controversy over the stochastic density functional equations","volume":"33","author":"Frusawa","year":"2000","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9325","DOI":"10.1088\/0305-4470\/37\/40\/001","article-title":"Dynamical density functional theory for interacting Brownian particles: Stochastic or deterministic?","volume":"37","author":"Archer","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"053032","DOI":"10.1088\/1367-2630\/16\/5\/053032","article-title":"Generalized Langevin equations for a driven tracer in dense soft colloids: Construction and applications","volume":"16","author":"Jacquin","year":"2014","journal-title":"New J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1137\/18M1172697","article-title":"A regularized Dean\u2013Kawasaki model: Derivation and analysis","volume":"51","author":"Cornalba","year":"2019","journal-title":"Siam J. Math. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5716","DOI":"10.1016\/j.physa.2008.06.016","article-title":"Hamiltonian and Brownian systems with long-range interactions: V. stochastic kinetic equations and theory of fluctuations","volume":"387","author":"Chavanis","year":"2008","journal-title":"Phys. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3205","DOI":"10.3390\/e17053205","article-title":"Generalized stochastic Fokker-Planck equations","volume":"17","author":"Chavanis","year":"2015","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chavanis, P.H. (2019). The Generalized Stochastic Smoluchowski Equation. Entropy, 21.","DOI":"10.3390\/e21101006"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"065003","DOI":"10.1088\/1751-8121\/aaf65c","article-title":"Stochastic dynamics and thermodynamics around a metastable state based on the linear Dean\u2013Kawasaki equation","volume":"52","author":"Frusawa","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"023106","DOI":"10.1088\/1742-5468\/2016\/02\/023106","article-title":"The conductivity of strong electrolytes from stochastic density functional theory","volume":"2016","author":"Dean","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"240602","DOI":"10.1103\/PhysRevLett.116.240602","article-title":"Nonequilibrium Tuning of the Thermal Casimir Effect","volume":"116","author":"Dean","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"118002","DOI":"10.1103\/PhysRevLett.118.118002","article-title":"Universal long ranged correlations in driven binary mixtures","volume":"118","author":"Poncet","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"134507","DOI":"10.1063\/1.4979659","article-title":"A Gaussian theory for fluctuations in simple liquids","volume":"146","author":"Dean","year":"2017","journal-title":"J. Chem. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"084503","DOI":"10.1063\/1.5019424","article-title":"Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory","volume":"148","author":"Solon","year":"2018","journal-title":"J. Chem. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3569","DOI":"10.1073\/pnas.1713573115","article-title":"Energy dissipation and fluctuations in a driven liquid","volume":"115","author":"Tociu","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1080\/00018737900101365","article-title":"The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids","volume":"28","author":"Evans","year":"1979","journal-title":"Adv. Phys."},{"key":"ref_29","unstructured":"Hansen, J.P., and McDonald, I.R. (2013). Theory of Simple Liquids, Academic Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"021503","DOI":"10.1103\/PhysRevE.75.021503","article-title":"Steric effects in the dynamics of electrolytes at large applied voltages. II. Modified Poisson-Nernst-Planck equations","volume":"75","author":"Kilic","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2680","DOI":"10.1063\/1.1682399","article-title":"Hypernetted chain solutions for the classical one-component plasma up to \u0393 = 7000","volume":"61","author":"Ng","year":"1974","journal-title":"J. Chem. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1435","DOI":"10.1103\/PhysRevE.59.1435","article-title":"Decay of correlations in fluids: The one-component plasma from Debye-H\u00fcckel to the asymptotic-high-density limit","volume":"59","author":"Evans","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"494206","DOI":"10.1088\/0953-8984\/20\/49\/494206","article-title":"Local molecular field theory for the treatment of electrostatics","volume":"20","author":"Rodgers","year":"2008","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/BF02124750","article-title":"On the Lambert W function","volume":"5","author":"Corless","year":"1996","journal-title":"Adv. Comput. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:45:41Z","timestamp":1760190341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/1\/34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,25]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["e22010034"],"URL":"https:\/\/doi.org\/10.3390\/e22010034","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,25]]}}}