Projection operators in statistical mechanics: a pedagogical approach

Abstract

The Mori-Zwanzig projection operator formalism is one of the central tools of nonequilibrium statistical mechanics, allowing to derive macroscopic equations of motion from the microscopic dynamics through a systematic coarse-graining procedure. It is important as a method in physical research and gives many insights into the general structure of nonequilibrium transport equations and the general procedure of microscopic derivations. Therefore, it is a valuable ingredient of basic and advanced courses in statistical mechanics. However, accessible introductions to this methodin particular in its more advanced forms -are extremely rare. In this article, we give a simple and systematic introduction to the Mori-Zwanzig formalism, which allows students to understand the methodology in the form it is used in current research. This includes both basic and modern versions of the theory. Moreover, we relate the formalism to more general aspects of statistical mechanics and quantum mechanics. Thereby, we explain how this method can be incorporated into a lecture course on statistical mechanics as a way to give a general introduction to the study of nonequilibrium systems. Applications, in particular to spin relaxation and dynamical density functional theory, are also discussed. * Corresponding author: raphael.wittkowski@uni-muenster.de 1 Active particles are particles that convert energy into directed motion. A good example are swimming microorganisms [1]. 2 This is not possible for active systems [7].

Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory

“…An alternative method used in this area are coarse-grained field theories, which describe in a simplified way the microscopic dynamics they are derived from on a larger scale. A paradigmatic example of such a theory is DDFT 18 , 19 . The results obtained from coarse-grained theories are often in excellent agreement with particle-based simulations and have a clear connection to the microscopic description, while providing additional analytical insights 20 .…”

“…DDFT can be extended to mixtures 37 , 38 , which makes it applicable to chemical reactions. Although DDFT is not an exact theory (it is based on the assumption that the density is the only slow variable in the system 18 , 19 ), it is nevertheless a significant improvement compared to the standard diffusion equation as it allows to incorporate the effects of particle interactions and generally shows excellent agreement with microscopic simulations. In particular, it allows to incorporate the effects of particle interactions such as crowding in reaction–diffusion equations.…”

“…DDFT describes the diffusive relaxation of an interacting system and is thus appropriate if we make the plausible approximation that the underlying diffusion behavior of persons is Markovian 62 and ergodic 63 . Using the Mori–Zwanzig formalism 64 – 66 , one can connect the DDFT model and its coefficients to the dynamics of the individual persons 18 , 19 . The extended model reads Note that we allow for different mobilities Γ S , Γ I , and Γ R for the different fields S , I , and R .…”

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

“…Various extensions of the Nakajima-Zwanzig operator technique [229] and the Mori-Zwanzig representation of a projected dynamical system have been proposed in the literature [230]. The idea of coupling the system with a heat bath, conventionally used in many classical settings, such as molecular dynamics, faces new challenges in the context of these problems.…”

Dynamic density correlations in a baryon rich fluid using Mori-Zwanzig projection operator method

“…One then introduces the projection operator which project onto subspace formed by the slow variables. While the Mori-Zwanzig formalism has been traditionally applied to non-relativistic systems [41,42], it has recently attracted a lot of attention in high energy physics [43][44][45][46] and cosmology research [47].…”