A tale of three approaches: dynamical phase transitions for weakly bound Brownian particles
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929432972754944 |
|---|---|
| author | Defaveri, Lucianno Barkai, Eli Kessler, David A. |
| author_facet | Defaveri, Lucianno Barkai, Eli Kessler, David A. |
| contents | We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as $V(x) \sim |x|^α$, with $0 < α< 1$. The probability density function $P(x,t)$ at long times reaches the Boltzmann-Gibbs equilibrium state, with all moments finite. However, the system's relaxation is not exponential, as is usual for a confining system with a well-defined equilibrium, but instead follows a stretched exponential $e^{- \mathrm{const} \, t^ν}$ with exponent $ν=α/(2+α)$. This problem is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for $P(x,t)$. This rate-function, which is an important tool from large deviation theory, also displays anomalous time scaling and a dynamical phase transition. Second, through the eigenfunctions of the Fokker-Planck operator, we obtain, using the WKB method, more complete solutions that reproduce the rate function approach. Finally, we show how the alternative path-integral formalism allows us to recover the same results, with the above rate-function being the solution of the classical Hamilton-Jacobi equation describing the most probable path. Properties such as parity, the role of initial conditions, and the dynamical phase transition are thoroughly studied in all three approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16598 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A tale of three approaches: dynamical phase transitions for weakly bound Brownian particles Defaveri, Lucianno Barkai, Eli Kessler, David A. Statistical Mechanics We investigate a system of Brownian particles weakly bound by attractive parity-symmetric potentials that grow at large distances as $V(x) \sim |x|^α$, with $0 < α< 1$. The probability density function $P(x,t)$ at long times reaches the Boltzmann-Gibbs equilibrium state, with all moments finite. However, the system's relaxation is not exponential, as is usual for a confining system with a well-defined equilibrium, but instead follows a stretched exponential $e^{- \mathrm{const} \, t^ν}$ with exponent $ν=α/(2+α)$. This problem is studied from three perspectives. First, we propose a straightforward and general scaling rate-function solution for $P(x,t)$. This rate-function, which is an important tool from large deviation theory, also displays anomalous time scaling and a dynamical phase transition. Second, through the eigenfunctions of the Fokker-Planck operator, we obtain, using the WKB method, more complete solutions that reproduce the rate function approach. Finally, we show how the alternative path-integral formalism allows us to recover the same results, with the above rate-function being the solution of the classical Hamilton-Jacobi equation describing the most probable path. Properties such as parity, the role of initial conditions, and the dynamical phase transition are thoroughly studied in all three approaches. |
| title | A tale of three approaches: dynamical phase transitions for weakly bound Brownian particles |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2407.16598 |