On the Mean Residence Time in Stochastic Lattice-Gas Models
Abstract
A heuristic law widely used in fluid dynamics for steady flows states that the amount of a fluid in a control volume is the product of the fluid influx and the mean time that the particles of the fluid spend in the volume, or mean residence time. We rigorously prove that if the mean residence time is introduced in terms of sample-path averages, then stochastic lattice-gas models with general injection, diffusion, and extraction dynamics verify this law. Only mild assumptions are needed in order to make the particles distinguishable so that their residence time can be unambiguously defined. We use our general result to obtain explicit expressions of the mean residence time for the Ising model on a ring with Glauber + Kawasaki dynamics and for the totally asymmetric simple exclusion process with open boundaries.
Keywords
Residence time Interacting particle systems Sample-path averages Strong law of large numbersMathematics Subject Classification
60F15 60J27 60K35 82C20 82C22References
- 1.van der Ent, R.J., Tuinenburg, O.A.: The residence time of water in the atmosphere revisited. Hydrol. Earth Syst. Sci. 21, 779–790 (2017)ADSCrossRefGoogle Scholar
- 2.Sincero, A.P., Sincero, G.A.: Physical-Chemical Treatment of Water and Wastewater. CRC Press, Boca Raton (2003)Google Scholar
- 3.Nauman, E.B.: Residence time theory. Ind. Eng. Chem. Res. 47, 3752–3766 (2008)CrossRefGoogle Scholar
- 4.Weiss, M.: The relevance of residence time theory to pharmacokinetics. Eur. J. Clin. Pharmacol. 43, 571–579 (1992)CrossRefGoogle Scholar
- 5.Zamparo, M., Valdembri, D., Serini, G., Kolokolov, I.V., Lebedev, V.V., DallAsta, L., Gamba, A.: Optimality in self-organized molecular sorting (in preparation)Google Scholar
- 6.Little, J.D.C.: Little’s law as viewed on its 50th anniversary. Oper. Res. 59, 536–549 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Griffeath, D.: Frank Spitzer’s pioneering work on interacting particle systems. Ann. Probab. 21, 608–621 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
- 9.Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Stochastic interacting particle systems out of equilibrium. J. Stat. Mech. 2007(07), P07014 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
- 11.Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A 26, 1493–1517 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 12.Schütz, G.M.: Exactly solvable models for many-body systems far from equilibrium. In: Domb, C., Lebowitz, J. (eds.) Phase Transitions and Critical Phenomena, vol. 19, pp. 1–251. Academic Press, San Diego (2001)CrossRefGoogle Scholar
- 13.Cirillo, E.N.M., Krehel, O., Muntean, A., van Santen, R., Sengar, A.: Residence time estimates for asymmetric simple exclusion dynamics on strips. Physica A 442, 436–457 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 14.Messelink, J., Rens, R., Vahabi, M., MacKintosh, F.C., Sharma, A.: On-site residence time in a driven diffusive system: violation and recovery of a mean-field description. Phys. Rev. E 93, 012119 (2016)ADSCrossRefGoogle Scholar
- 15.Norris, J.R.: Markov Chains, reprinted edn. Cambridge University Press, Cambridge (1998)Google Scholar
- 16.Serfozo, R.: Basics of Applied Stochastic Processes. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
- 17.Presutti, E.: Scaling Limits in Statistical Mechanics and Microstructures in Continuum Mechanics. Springer, Berlin (2009)zbMATHGoogle Scholar