Stochastic mechanics in the context of the properties of living systems (original) (raw)

Statistical mechanics in a nutshell

Choice Reviews Online, 2012

Introduction l 1.1 The Subject Matter of Statistical Mechanics 1.2 Statistical Postulates 1.3 An Example: The Ideal Gas 1.4 Conclusions Recommended Reading 2~| Thermodynamics 2.1 Thermodynamic Systems 2.2 Extensive Variables 2.3 The Central Problem of Thermodynamics 2.4 Entropy 2.5 Simple Problems 2.6 Heat and Work 2.7 The Fundamental Equation 2.8 Energy Scheme 2.9 Intensive Variables and Thermodynamic Potentials 2.10 Free Energy and Maxwell Relations 2.11 Gibbs Free Energy and Enthalpy 2.12 The Measure of Chemical Potential 2.13 The Koenig Born Diagram Contents 2.14 Other Thermodynamic Potentials 36 2.15 The Euler and Gibbs-Duhem Equations 37 2.16 Magnetic Systems 39 2.17 Equations of State 2.18 Stability 41 2.19 Chemical Reactions 44 2.20 Phase Coexistence 45 2.21 The Clausius-Clapeyron Equation 47 2.22 The Coexistence Curve 48

STATISTICAL MECHANICS

132 STATISTICAL MECHANICS where w ( E ) is called the density of states of the system at the energy E and is defined by

What is life? A perspective of the mathematical kinetic theory of active particles

Mathematical Models and Methods in Applied Sciences

The modeling of living systems composed of many interacting entities is treated in this paper with the aim of describing their collective behaviors. The mathematical approach is developed within the general framework of the kinetic theory of active particles. The presentation is in three parts. First, we derive the mathematical tools, subsequently, we show how the method can be applied to a number of case studies related to well defined living systems, and finally, we look ahead to research perspectives.

Principles of Statistical Mechanics

Molecular Dynamics Simulations in Statistical Physics: Theory and Applications, 2020

In this chapter, we will describe some fundamental topics of thermodynamics and statistical mechanics. Furthermore, we will discuss the energy or particle number fluctuations in different statistical ensembles and their differences. For further reading on the statistical mechanics, one should consider the books by Gibbs (1902), Hansen and McDonald (1986), and McQuarrie (1976, 2000). 3.1 Systems In statistical mechanics systems play the same role as particles in kinetic theory. The system has a very general concept in statistical mechanics, and it may include any physical object. For example, we can mention the galaxy, a planet, crystal and its fundamental mode of vibration, an atom in a crystal, an electron of the atom, and a quantum state in which that electron could reside. Statistical mechanics pays special attention to systems that couple only weakly to the rest of the universe. With other words, in statistical mechanics, the focus is the systems whose relevant internal evolution timescales, τ int , are short compared with the external timescales, τ ext , on which they exchange energy, entropy, particles, and so on, with their surrounding environments. These systems are also called semiclosed. In contrast, a system for which in the idealized limit external interactions are completely ignored, is called closed system. The statistical mechanics formalism for dealing with closed systems relies on the assumption τ int /τ ext 1. Therefore, it depends on the two length scale expansions, τ int and τ ext. If a semi-closed classical system does not interact with the external universe, then it is considered a closed system, and Hamiltonian dynamics (Poole 2001) describe its time evolution. In this textbook we are discussing the classical systems, therefore,

On the mathematical theory of living systems II: The interplay between mathematics and system biology

Computers & Mathematics With Applications, 2011

This paper aims at showing how the so-called mathematical kinetic theory for active particles can be properly developed to propose a new system biology approach. The investigation begins with an analysis of complexity in biological systems, continues with reviewing a general methodology to reduce complexity and furnishes the mathematical tools to describe the time evolution of such systems by capturing all their features.

What is Statistical Mechanics?

Thermodynamics describes a large class of phenomena we observe in macroscopic systems. The aim of statistical mechanics is to account for this behaviour in terms of the dynamical laws governing the microscopic constituents of macroscopic systems and probabilistic assumptions. This article provides a survey of the discussion about the foundation of statistical mechanics by introducing the basic approaches and discussing their merits as well as their problems. After a brief review of classical mechanics, which provides the background against which statistical mechanics is formulated, we discuss the two main theoretical approaches to statistical mechanics, one of which can be associated with Boltzmann and the other with Gibbs. We end with a discussion of remaining issues and open questions.

Statistical Physics of Evolving Systems

Entropy

Evolution is customarily perceived as a biological process. However, when formulated in terms of physics, evolution is understood to entail everything. Based on the axiom of everything comprising quanta of actions (e.g., quanta of light), statistical physics describes any system evolving toward thermodynamic balance with its surroundings systems. Fluxes of quanta naturally select those processes leveling out differences in energy as soon as possible. This least-time maxim results in ubiquitous patterns (i.e., power laws, approximating sigmoidal cumulative curves of skewed distributions, oscillations, and even the regularity of chaos). While the equation of evolution can be written exactly, it cannot be solved exactly. Variables are inseparable since motions consume driving forces that affect motions (and so on). Thus, evolution is inherently a non-deterministic process. Yet, the future is not all arbitrary but teleological, the final cause being the least-time free energy consumptio...

A Field Guide to Recent Work on the Foundations of Statistical Mechanics

2008

This is an extensive review of recent work on the foundations of statistical mechanics. Subject matters discussed include: interpretation of probability, typicality, recurrence, reversibility, ergodicity, mixing, coarse graining, past hypothesis, reductionism, phase average, thermodynamic limit, interventionism, entropy.

The Significance of Non-ergodic Property of Statistical Mechanics Systems for Understanding Resting State of a Living Cell

2011

A better grasp of the physical foundations of life is necessary before we can understand the processes occurring inside a living cell. In his physical theory of the cell, American physiologist Gilbert Ling introduced an important notion of the resting state of the cell. He describes this state as an independent stable thermodynamic state of a living substance in which it has stored all the energy it needs to perform all kinds of biological work. This state is characterized by lower entropy of the system than in an active state. The main contribution to this reduction in entropy is made by the cellular water (the dominant component with a concentration of 14 M) which remains in a bound quasi-crystallized state in a resting cell. When the cell becomes active the water gets desorbed and the system’s entropy goes up sharply while the free energy of the system decreases as it is used up for biological work. However, Ling’s approach is primarily qualitative in terms ofh thermodynamics and it needs to be characterized more specifically. To this end, we propose a new thermodynamic approach to studying Ling’s model of the living cell (Ling’s cell), the centrepiece off which is the non-ergodicity property which has recently been proved for a wide range of systems in statistical mechanics (Prokhorenko, 2009). In many ways this new thermodynamics overlaps with the standard quasi-stationary thermodynamics and is therefore compatible with the principles of the Ling cell, however a number of new specific results take into account the existence of several non-trivial motion integrals communicating with each other, whose existence follows from the nonergodicity of the system (Ling’s cell). These results allowed us to develop general thermodynamic approaches to explaining some of the well-known physiological phenomena, which can be used for further physical analysis of these phenomena using specific physical models.