Diffusion of finitesized hardcore interacting particles in a onedimensional box: Tagged particle dynamics
(2009) In Physical Review E (Statistical, Nonlinear, and Soft Matter Physics) 80(5). Abstract
 We solve a nonequilibrium statisticalmechanics problem exactly, namely, the singlefile dynamics of N hardcore interacting particles (the particles cannot pass each other) of size Delta diffusing in a onedimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rho T(yT,t vertical bar yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the Nparticle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T, we arrive at an exact expression for rho T(yT,t... (More)
 We solve a nonequilibrium statisticalmechanics problem exactly, namely, the singlefile dynamics of N hardcore interacting particles (the particles cannot pass each other) of size Delta diffusing in a onedimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rho T(yT,t vertical bar yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the Nparticle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T, we arrive at an exact expression for rho T(yT,t vertical bar yT,0) in terms of Jacobi polynomials or hypergeometric functions. Going beyond previous studies, we consider the asymptotic limit of large N, maintaining L finite, using a nonstandard asymptotic technique. We derive an exact expression for rho T(yT,t vertical bar yT,0) for a tagged particle located roughly in the middle of the system, from which we find that there are three time regimes of interest for finitesized systems: (A) for times much smaller than the collision time t <tau(coll)=1/(rho D2), where rho=N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion; (B) for times much larger than the collision time t tau(coll) but times smaller than the equilibrium time t <tau(eq)=L2/D, we find a singlefile regime where rho T(yT,t vertical bar yT,0) is a Gaussian with a meansquare displacement scaling as t(1/2); and (C) for times longer than the equilibrium time t tau(eq), rho T(yT,t vertical bar yT,0) approaches a polynomialtype equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1532986
 author
 Lizana, L. and Ambjörnsson, Tobias ^{LU}
 organization
 publishing date
 2009
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 statistical, probability, polynomials, diffusion, mechanics, manybody problems
 in
 Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
 volume
 80
 issue
 5
 publisher
 American Physical Society
 external identifiers

 wos:000272309500009
 scopus:70449687989
 pmid:20364943
 ISSN
 15393755
 DOI
 10.1103/PhysRevE.80.051103
 language
 English
 LU publication?
 yes
 id
 a63884bc2845430bb0e18de611f0faeb (old id 1532986)
 date added to LUP
 20160401 12:30:10
 date last changed
 20201208 01:16:00
@article{a63884bc2845430bb0e18de611f0faeb, abstract = {We solve a nonequilibrium statisticalmechanics problem exactly, namely, the singlefile dynamics of N hardcore interacting particles (the particles cannot pass each other) of size Delta diffusing in a onedimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rho T(yT,t vertical bar yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the Nparticle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T, we arrive at an exact expression for rho T(yT,t vertical bar yT,0) in terms of Jacobi polynomials or hypergeometric functions. Going beyond previous studies, we consider the asymptotic limit of large N, maintaining L finite, using a nonstandard asymptotic technique. We derive an exact expression for rho T(yT,t vertical bar yT,0) for a tagged particle located roughly in the middle of the system, from which we find that there are three time regimes of interest for finitesized systems: (A) for times much smaller than the collision time t <tau(coll)=1/(rho D2), where rho=N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion; (B) for times much larger than the collision time t tau(coll) but times smaller than the equilibrium time t <tau(eq)=L2/D, we find a singlefile regime where rho T(yT,t vertical bar yT,0) is a Gaussian with a meansquare displacement scaling as t(1/2); and (C) for times longer than the equilibrium time t tau(eq), rho T(yT,t vertical bar yT,0) approaches a polynomialtype equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems.}, author = {Lizana, L. and Ambjörnsson, Tobias}, issn = {15393755}, language = {eng}, number = {5}, publisher = {American Physical Society}, series = {Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)}, title = {Diffusion of finitesized hardcore interacting particles in a onedimensional box: Tagged particle dynamics}, url = {http://dx.doi.org/10.1103/PhysRevE.80.051103}, doi = {10.1103/PhysRevE.80.051103}, volume = {80}, year = {2009}, }