Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics
We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Δ diffusing in a one-dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2011
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| Online Access: | http://hdl.handle.net/1721.1/65085 |
| _version_ | 1826190249562210304 |
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| author | Lizana, L. Ambjornsson, T. |
| author2 | Massachusetts Institute of Technology. Department of Chemistry |
| author_facet | Massachusetts Institute of Technology. Department of Chemistry Lizana, L. Ambjornsson, T. |
| author_sort | Lizana, L. |
| collection | MIT |
| description | We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Δ diffusing in a one-dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function ρT(yT,t∣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 N-particle 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 ρT(yT,t∣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 ρT(yT,t∣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 finite-sized systems: (A) for times much smaller than the collision time t«τcoll=1/(ϱ2D), where ϱ=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«τcoll but times smaller than the equilibrium time t«τeq=L2/D, we find a single-file regime where ρT(yT,t∣yT,0) is a Gaussian with a mean-square displacement scaling as t1/2; and (C) for times longer than the equilibrium time t«τeq, ρT(yT,t∣yT,0) approaches a polynomial-type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems. |
| first_indexed | 2024-09-23T08:37:21Z |
| format | Article |
| id | mit-1721.1/65085 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:37:21Z |
| publishDate | 2011 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/650852022-09-23T13:20:58Z Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics Lizana, L. Ambjornsson, T. Massachusetts Institute of Technology. Department of Chemistry Ambjornsson, T. Ambjornsson, T. We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Δ diffusing in a one-dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function ρT(yT,t∣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 N-particle 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 ρT(yT,t∣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 ρT(yT,t∣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 finite-sized systems: (A) for times much smaller than the collision time t«τcoll=1/(ϱ2D), where ϱ=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«τcoll but times smaller than the equilibrium time t«τeq=L2/D, we find a single-file regime where ρT(yT,t∣yT,0) is a Gaussian with a mean-square displacement scaling as t1/2; and (C) for times longer than the equilibrium time t«τeq, ρT(yT,t∣yT,0) approaches a polynomial-type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems. Danish National Research Foundation Knut and Alice Wallenberg Foundation 2011-08-04T21:15:42Z 2011-08-04T21:15:42Z 2009-11 2009-07 Article http://purl.org/eprint/type/JournalArticle 1539-3755 1550-2376 http://hdl.handle.net/1721.1/65085 Lizana, L., and T. Ambjörnsson. “Diffusion of Finite-sized Hard-core Interacting Particles in a One-dimensional Box: Tagged Particle Dynamics.” Physical Review E 80.5 (2009) : 051103. © 2009 The American Physical Society en_US http://dx.doi.org/10.1103/PhysRevE.80.051103 Physical Review E Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society APS |
| spellingShingle | Lizana, L. Ambjornsson, T. Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title | Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title_full | Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title_fullStr | Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title_full_unstemmed | Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title_short | Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics |
| title_sort | diffusion of finite sized hard core interacting particles in a one dimensional box tagged particle dynamics |
| url | http://hdl.handle.net/1721.1/65085 |
| work_keys_str_mv | AT lizanal diffusionoffinitesizedhardcoreinteractingparticlesinaonedimensionalboxtaggedparticledynamics AT ambjornssont diffusionoffinitesizedhardcoreinteractingparticlesinaonedimensionalboxtaggedparticledynamics |