A microscopic, non-equilibrium, statistical field theory for cosmic structure formation
Matthias Bartelmann, Felix Fabis, Daniel Berg, Elena Kozlikin, Robert Lilow, Celia Viermann
备注：
48 pages, 5 figures; accepted for publication in the New Journal of Physics; upon referees' requests, combines earlier submissions 1411.0806, 1411.1153 and 1411.1502
Building upon the recent pioneering work by Mazenko and by Das and Mazenko, we develop a microscopic, non-equilibrium, statistical field theory for initially correlated canonical ensembles of classical microscopic particles obeying Hamiltonian dynamics. Our primary target is cosmic structure formation, where initial Gaussian correlations in phase space are believed to be set by inflation. We give an exact expression for the generating functional of this theory and work out suitable approximations. We specify the initial correlations by a power spectrum and derive general expressions for the correlators of the density and the response field. We derive simple closed expressions for the lowest-order contributions to the non-linear cosmological power spectrum, valid for arbitrary wave numbers. We further calculate the bispectrum expected in this theory within these approximations and the power spectrum of cosmic density fluctuations to first order in the gravitational interaction, using a recent improvement of the Zel'dovich approximation. We show that, with a modification motivated by the adhesion approximation, the non-linear growth of the density power spectrum found in numerical simulations of cosmic structure evolution is reproduced well to redshift zero and for arbitrary wave numbers even within first-order perturbation theory. Our results present the first fully analytic calculation of the non-linear power spectrum of cosmic structures.
The distribution function P(j) of the microscopic energy flux, j, in equilibrium state is studied. It is observed that P(j) has a broad peak in small j regime and a stretched-exponential decay for large j. The peak structure originates in a potential advection term and energy transfer term between the particles. The stretched exponential tail comes from the momentum energy advection term.
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Quantum master equations form an important tool in the description of transport problems in open quantum systems. However, they suffer from the difficulty that the shape of the Lindblad dissipator depends sensibly on the system Hamiltonian. Consequently, most of the work done in this field has focused on phenomenological dissipators which act locally on different parts of the system. In this paper we show how to construct Lindblad dissipators for quantum many-body systems starting from a microscopic theory of the system-bath interaction. We consider specifically a one-dimensional bosonic tight…
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Recently Mazenko and Das and Mazenko introduced a non-equilibrium field theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the m…
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The dynamics of free vibrations of a chain of atoms is investigated taking into account the retardation of interactions. It is shown that all oscillations of the circuit are damped. The dynamics of forced vibrations of this chain of atoms is investigated. It is shown that, regardless of the initial conditions, the system passes into a stationary state of dynamic equilibrium with an external field, which depends both on the properties of the system and on the parameters of the external field. A non-statistical dynamic mechanism of the process of irreversible establishment of the state of thermo…
The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results which have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.
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In this article, we review the current state of the art in using levitated nanoparticles to answer questions related to thermodynamics and non-equilibrium physics. We begin in Section 2 with a summary of the relevant deterministic and stochastic forces, which determine the particle dynamics and allow for control of the particle. In Section 3 we give a brief review of the stochastic (i.e. Brownian) motion of levitated particles, since the Brownian particle is fundamental to the theory of stochastic thermodynamics. Then we discuss the stability of the trapped particles and the related Kramers es…
Thermal contact is the archetype of non-equilibrium processes driven by constant non-equilibrium constraints when the latter are enforced by reservoirs exchanging conserved microscopic quantities. At a mesoscopic scale only the energies of the macroscopic bodies are accessible together with the configurations of the contact system. We consider a class of models where the contact system, as well as macroscopic bodies, have a finite number of possible configurations. The global system with only discrete degrees of freedom has no microscopic Hamiltonian dynamics, but it is shown that, if the micr…
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The pressure tensor (equivalent to the negative stress tensor) at both microscopic and macroscopic levels is fundamental to many aspects of engineering and science, including fluid dynamics, solid mechanics, biophysics, and thermodynamics. In this perspective paper, we review methods to calculate the microscopic pressure tensor. Connections between different pressure forms for equilibrium and non-equilibrium systems are established. We also point out several challenges in the field, including the historical controversies over the definition of the microscopic pressure tensor; the difficulties …
In recent years statistical physicists have developed {\it discrete} "particle-hopping" models of vehicular traffic, usually formulated in terms of {\it cellular automata}, which are similar to the microscopic models of interacting charged particles in the presence of an external electric field. Concepts and techniques of non-equilibrium statistical mechanics are being used to understand the nature of the steady states and fluctuations in these so-called "microscopic" models. In this brief review we explain, primarily to the nonexperts, these models and the physical implications of the results…
Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on projection operator techniques such as the Mori-Zwanzig (MZ) formalism or by "integrating out" the bath degrees of freedom. Based on exact analytical results we show that both routes can lead to fundamentally different GLEs and that the origin of these differences is based inherently on the non-equilibrium nature of the microscopic stochastic model. The most important conceptional difference between the …
We investigate the microscopic features of bosonic quantum transport in a non-equilibrium steady state, which breaks time reversal invariance spontaneously. The analysis is based on the probability distributions, generated by the correlation functions of the particle current and the entropy production operator. The general approach is applied to an exactly solvable model with a point-like interaction driving the system away from equilibrium. The quantum fluctuations of the particle current and the entropy production are explicitly evaluated in the zero frequency limit. It is shown that all mom…
It was shown in [PRL 114, 138301 (2015)] that a remarkably simple dynamical model exhibits many of the complex flow regimes and non-equilibrium phase transitions characteristic of complex fluids. By removing extraneous detail, this simplest microscopic model of non-Newtonian flow can reveal the universal physics relevant to all complex fluids. Here we present more detailed results and a full derivation of the model's compact mean-field constitutive relation, with great potential scope for insights into universality and tractable mathematics. By enforcing local conservation of angular momentum,…
Dense particle packings acquire rigidity through a nonequilibrium jamming transition commonly observed in materials from emulsions to sandpiles. We describe athermal packings and their observed geometric phase transitions using fully equilibrium statistical mechanics and develop a microscopic many-body mean-field theory of the jamming transition for soft repulsive spherical particles. We derive analytically some of the scaling laws and exponents characterizing the transition and obtain predictions for microscopic correlation functions of jammed states that are amenable to experimental verifica…
Molecular dynamics simulations allow to study the structure and dynamics of single biomolecules in microscopic detail. However, many processes occur on time scales beyond the reach of fully atomistic simulations and require coarse-grained multiscale models. While systematic approaches to construct such models have become available, these typically rely on microscopic dynamics that obey detailed balance. In vivo, however, biomolecules are constantly driven away from equilibrium in order to perform specific functions and thus break detailed balance. Here we introduce a method to construct Markov…
Macroscopic thermodynamics of equilibrium is constructed for systems obeying power-law canonical distributions. With this, the connection between macroscopic thermodynamics and microscopic statistical thermodynamics is generalized. This is complementary to the Gibbs theorem for the celebrated exponential canonical distributions of systems in contact with a heat bath. Thereby, a thermodynamic basis is provided for power-law phenomena ubiquitous in nature.
This paper provides an introduction to some stochastic models of lattice gases out of equilibrium and a discussion of results of various kinds obtained in recent years. Although these models are different in their microscopic features, a unified picture is emerging at the macroscopic level, applicable, in our view, to real phenomena where diffusion is the dominating physical mechanism. We rely mainly on an approach developed by the authors based on the study of dynamical large fluctuations in stationary states of open systems. The outcome of this approach is a theory connecting the non equilib…
For classical discrete systems under constant composition, statistical mechanics tells us that a set of microscopic state dominantly contributing to thermodynamically equilibrium state should depend on temperature as well as on many-body interaction (i.e. thermodynamic information), through Boltzamann factor of exp(-bE). Despite this fact, our recent study reveals that a single (and a few additional) microscopic state (called projection state: PS), whose structure can be known a priori without r equiring thermodynamic information, can universally characterize equiibrium properties for disorder…
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We use surface tension to distinguish between phases with isotropic internal structure from phases which are microscopically anisotropic. There are many interesting open problems, especially in two dimensions, and in phase coexistence.
Measurements of any property of a microscopic system are bound to show significant deviations from the average, due to thermal fluctuations. For time-integrated currents such as heat, work or entropy production in a steady state, it is in fact known that there will be long stretches of fluctuations both above as well as below the average, occurring equally likely at large times. In this paper we show that for any finite-time measurement in a non-equilibrium steady state - rather counter-intuitively - fluctuations below the average are more probable. This discrepancy is higher when the system i…
The most complicated phenomena of equilibrium statistics, phase separations and transitions of various order and critical phenomena, can clearly and sharply be seen even for small systems in the topology of the curvature of the microcanonical entropy $S_{B}(E,N)=\ln[W(E,N)]$ as function of the conserved energy, particle number etc.. Also the equilibrium of the largest possible interacting many-body systems like self-gravitating systems is described by the topology of the entropy surface $S_{B}(E,N,L)$ where $L$ is the angular momentum. Conventional (canonical) statistical mechanics describes p…
We investigate non-equilibrium nature of fluctuations of black hole horizons by applying the fluctuation theorems and the Jarzynski equality developed in the non-equilibrium statistical physics. These theorems applied to space-times with black hole horizons lead to the generalized second law of thermodynamics. It is also suggested that the second law should be violated microscopically so as to satisfy the Jarzynski equality.
Finite-Size Effects in Quantum Metrology at Strong Coupling: Microscopic vs Phenomenological Approaches
Koç University（科奇大学）；Korea Research Institute of Standards and Science（韩国标准科学研究所）；University of Calgary（卡尔加里大学）；TÜBİTAK Research Institute for Fundamental Sciences (TBAE)（土耳其科技研究基金会基础科学研究院（TBAE））
We describe an extension of the Keldysh method for fermions from constant temperature to steady state case with spatially varying temperature field.This is done with the use on the imaginary section of the Keldysh path of a thermal Hamiltonian obtained from the zero temperature relativistic Hamiltonian by coupling its density multiplicatively to the temperature field. We show that the two Hamiltonians commute, provided appropriate boundary conditions are imposed. A microscopical equation on the temperature field and the corresponding microscopical Fourier law of heat transfer are derived for t…
