Improved Field Theoretical Approach to Noninteracting Brownian Particles in a Quenched Random Potential
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Improved Field Theoretical Approach to Noninteracting Brownian Particles in a Quenched Random Potential Wonsang Lee Department of Physics, Konkuk University, Seoul 05029, Korea
Joonhyun Yeo
∗
Department of Physics, Konkuk University, Seoul 05029, Korea and School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea (Received 16 July 2020; revised 8 August 2020; accepted 10 August 2020) We construct a dynamical field theory for noninteracting Brownian particles in the presence of a quenched Gaussian random potential. The main variable for the field theory is the density fluctuation, which measures the difference between the local density and its average value. The average density is spatially inhomogeneous for a given realization of the random potential. It becomes uniform only after being averaged over the disorder configurations. We develop a diagrammatic perturbation theory for the density correlation function and calculate the zero-frequency component of the response function exactly by summing all the diagrams contributing to it. From this exact result and the fluctuation dissipation relation, which holds in equilibrium dynamics, we find that the connected density correlation function always decays to zero in the long-time limit for all values of disorder strength implying that the system always remains ergodic. This nonperturbative calculation relies on the simple diagrammatic structure of the present field theoretical scheme. We compare in detail our diagrammatic perturbation theory with the one used in a recent paper [B. Kim, M. Fuchs and V. Krakoviack, J. Stat. Mech. 2020, 023301 (2020)], which uses the density fluctuation around the uniform average, and discuss the difference in the diagrammatic structures of the two formulations. Keywords: Diffusion in random media, Brownian motion, Ergodicity breaking DOI: 10.3938/jkps.77.719
I. INTRODUCTION The dynamics of fluids in a quenched random environment has been studied in connection with many different research areas ranging from the structural glass transition [1–5] to biological [6,7] and engineering [8–11] applications. Theoretically, the main focus has been on the possible existence of an anomalous diffusion [12–22], which has been studied in connection with the spatial dimension and the range of the random potential and the thermal noise [23–25]. One of the main physical quantities is the late-time diffusion constant of a tagged particle. The calculation of effective transport properties in the presence of disorder has also been extensively studied [26–32]. All these studies are, however, based on the singleparticle picture. An alternative way is to use the field theoretical approaches [33–44], which are based on the Martin-Siggia-Rose-Janssen-de Dominicis (MSRJD)type dynamical field theory [45–47] for the stochastic ∗ E-mail:
equations governing collective variables such as the density field. using the field theoretical formalism has a number of advantages. One can, for example, extract physical information from the symmetry property of the action function
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