Statistical mechanics hydrodynamics

Dyson-type equations have been used extensively in quantum electrodynamics, quantum field theory, statistical mechanics, hydrodynamic instability and turbulent diffusion studies, and in investigations of electromagnetic wave propagation in a medium having a random refractive index (Tatarski, 1961). Also, this technique has recently been employed to study laser light scattering from a macromolecular solution in an electric field. 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

There are several attractive features of such a mesoscopic description. Because the dynamics is simple, it is both easy and efficient to simulate. The equations of motion are easily written and the techniques of nonequilibriun statistical mechanics can be used to derive macroscopic laws and correlation function expressions for the transport properties. Accurate analytical expressions for the transport coefficient can be derived. The mesoscopic description can be combined with full molecular dynamics in order to describe the properties of solute species, such as polymers or colloids, in solution. Because all of the conservation laws are satisfied, hydrodynamic interactions, which play an important role in the dynamical properties of such systems, are automatically taken into account. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

In addition to the general problem of the kinetics of the approach towards equilibrium, the statistical mechanics of irreversible phenomena concern in particular the study of transport phenomena. The latter are calculated in a stationary or quasi-stationary form (the distribution functions do not vary or vary in hydrodynamic fashion). Therefore, let us consider (see, for... 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

A major preoccupation of nonequilibrium statistical mechanics is to justify the existence of the hydrodynamic modes from the microscopic Hamiltonian dynamics. Boltzmann equation is based on approximations valid for dilute fluids such as the Stosszahlansatz. In the context of Boltzmann s theory, the concept of hydrodynamic modes has a limited validity because of this approximation. We may wonder if they can be justified directly from the microscopic dynamics without any approximation. If this were the case, this would be great progress... 

A major preoccupation in nonequilibrium statistical mechanics is to derive hydrodynamics and nonequilibrium thermodynamics from the microscopic Hamiltonian dynamics of the particles composing matter. The positions raYl= and momenta PaY i= of these particles obey Newton s equations or, equivalently, Hamilton s equations ... 

The Fisher relation (38) has a structure similar to a fluctuation dissipation relation in statistical mechanics It relates a macroscopic transport coefficient, the hydrodynamic speed, to the diffusion tensor and to the statistical properties of... 

The remaining four chapters discuss theoretical approaches and considerations which have been suggested to include the effects of many-body complications, to use approximate techniques, to use more realistic continuum hydrodynamic equations than the diffusion equation, and to use more satisfactory statistical mechanical descriptions of liquid structure. This work is still in a comparatively early stage of its development. There is a growing need for more detailed experiments which might probe the effects anticipated by these studies. 

It is the main purpose of the following articles by V. Enkelmann, H. Bassler and H. Sixl to review the present status of polydiacetylene research from the point of view of structure and reactivity including all the details known on the mechanism of polymerization of various diacetylenes. The material science aspects will not be treated to the same depth with exception of the photopolymerization and its possible application in the contribution of H. Bassler. Similarly, the solution properties of polydiacetylenes are not touched upon. The interested reader is refered to the current literature where the problems encountered when studying the solutions and the recrystallization behaviour of polyconjugated macromolecules have just started to be discussed adding a new chapter to the statistical mechanics and hydrodynamic behaviour of macromolecules... 

J. H. Irving and J. G. Kirkwood, The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics. J. Chem. 

Simple Models. Simulations are usually used for the direct calculation of properties or as an aid in the understanding of physical or chemical phenomena. However, they are also often carried as an aid in the development of simple models for future studies. This is particularly evident in the study of adsorption and flow in microporous systems, where standard hydrodynamic theories are inadequate but can in some cases be extended to treat the effects due to the confinement. Typically simulations of nanosystems need to be on longer timescales than those in the bulk due to the inhomogeneity of the system. Thus development of efficient models is important, and there has therefore been much activity in this field in recent years. Theories such as density functional theories have been extended and verified using simulation methods and simple statistical mechanical models have also been developed. 

W. Y. Zhang and R. Balescu (1988) Statistical-mechanics of a spin-polarized plasma. J. Plasma Phys. 40, pp. 199-213 ibid. (1988) Kinetic-equation, spin hydrodynamics and collisional depolarization rate in a spin-polarized plasma. J. Plasma Phys. 40, pp. 215-234... 

A wonderful article on random walks that gives a broad coverage of this important subject is On the wonderful world of random walks by E. W. Montroll and M. F. Shlesinger, in Studies in Statistical Mechanics, Volume XI, Elsevier Science Publishers, Amsterdam The Netherlands, 1984. This book goes under the title Nonequilibrium Phenomena II From Stochastics to Hydrodynamics, edited by J. L. Lebowitz and E. W. Montroll. 

Without doing detailed quantitative analysis of the data, it can be stated that the polyion diffusion can be qualitatively described by two theoretical concepts. The first concept capable of qualitative explanation of the polyion diffusion is the concept based on considering polyions as interacting Brownian particles with direct interactions between polyions and hydrodynamic interactions. The short-time collective diffusion coefficient for a system of interacting Brownian particles treated by statistical mechanics is calculated from the first cumulant F of the dynamic structure factor S(q, t) as [15-17]... 

Whereas in the previous chapter we have given a statistical-mechanical derivation of the Maxwell equations, and therefore discussed the influence of matter on the equations governing the average electric and magnetic fields, we shall be concerned in this chapter with the influence of the fields on the equation of motion (hydrodynamic equation). 

Formulation. In this section, we will summarize the procedure for calculating dynamic mechanical properties of dilute polymer solutions by applying statistical mechanics to the model mentioned above. Basic equations of the theory consist of equations of motion for the polymer elements, an equation of motion for the fluid (hydrodynamics), a diffusion equation to describe the statistical nature of the problem and an equation of stress. 

J. Andrew McCammon holds the Joseph E. Mayer Chair of Theoretical Chemistry at the University of California, San Diego (UCSD), and is an Investigator of the Howard Hughes Medical Institute. He received his Ph.D. in chemical physics in 1976 from Harvard University, where he worked with John Deutch on biological applications of statistical mechanics and hydrodynamics. In 1976-1978, he was a Research Fellow at Harvard, where he developed the computer simulation approach to protein dynamics in collaboration with Martin Karplus. He was an Assistant Professor and then M.D. Anderson Professor (1981-1994) at the University of Houston before moving to UCSD. He recalls with pleasure the first views of atomic dynamics in a protein molecule, generated slowly on a pen plotter during his postdoctoral work. 

Leimard-Jones, J.E., in "Fowler s Statistical Mechanics", Cambridge, 1936 Levich, V.G., Physico-Chemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.Y., 1962 Li, D., Gaydos, J. and Neumaim, A.W., Langmuir, 5(1989)1133 Lucassen-Reynders, E.H., Progr. Surface Membrane Sic., 3(1976)253 Lucassen-Reynders, E.H., Adsorption at Fluid Interfaces, in "Surfactant Science Series", Marcel Dekker, New York, 11(1981)... 

Chapter 11 reviews the statistical mechanical basis of hydrodynamics and discusses theories that may be used to extend hydrodynamics beyond the classical equations discussed in Chapter 10. Chapter 12 applies the statistical mechanical theory to the calculation of depolarized light-scattering spectra from dense liquids where interactions between anisotropic molecules are important. 

In this chapter, we begin with some remarks on the technological and scientific importance of complex materials and interfaces and motivate the study of interface and surface properties. We then review some of the physical and mathematical methods that are used in the subsequent discussions of interface and membrane statistical thermodynamics. Many of these topics are discussed more fully in the references and throughout this chapter. We begin with a review of classical statistical mechanics ", including a description of fluctuations about equilibrium and of binary mixtures. The mathematical description of an interface is then presented (using only vector calculus) and the calculation of the area and curvature of an interface wifli an arbitrary shape is demonstrated. Finally, the chapter is concluded by a brief summary of hydrodynamics. ... 

What are the chances that one or another theoretical study wiU be a success As history shows, it greatly depends on whether theorists can think of a nice, manageable model idealizing the real world. Of course, there are no ideally simple systems in nature. However, we can use our imagination and invent an ideal gas (whose molecules do not interact at all), an ideal crystal (with no defects at all to the regular atomic structure), and so on. As a matter of fact, you can say that all these models are ideal indeed, meaning that they are the best for physicists. This is because they are the simplest — but they are simultaneously the most basic ones. So one has to master them first, before moving any further in either statistical mechanics, or hydrodynamics, or solid state physics, or whatever chapter of physics. 

As shown in Figure 26.1, the wide gap opens up between the particle and continuum paradigms. This gap cannot be spanned using statistical mechanical methods only. The existing theoretical models to be applied in the mesoscale are based on heuristics obtained via downscaling of macroscopic models and upscaling particle approach. Simphfied theoretical models of complex fluid flows, e.g., flows in porous media, non-Newtonian fluid dynamics, thin film behavior, flows in presence of chemical reactions, and hydrodynamic instabilities formation, involve not only vah-dation but should be supported by more accurate computational models as well. However, until now, there has not been any precisely defined computational model, which operates in the mesoscale, in the range from 10 A to tens of microns. 

The cut-off radius rc t is defined arbitrarily and reveals the range of interaction between the fluid particles. DPD model with longer cut-off radius reproduces better dynamical properties of realistic fluids expressed in terms of velocity correlation function [80]. Simultaneously, for a shorter cut-off radius, the efficiency of DPD codes increases as 0(1 /t ut). which allows for more precise computation of thermodynamic properties of the particle system from statistical mechanics point of view. A strong background drawn from statistical mechanics has been provided to DPD [43,80,81] from which explicit formulas for transport coefficients in terms of the particle interactions can be derived. The kinetic theory for standard hydrodynamic behavior in the DPD model was developed by Marsh et al. [81] for the low-friction (small value of yin Equation (26.25)), low-density case and vanishing conservative interactions Fc. In this weak scattering theory, the interactions between the dissipative particles produce only small deflections. 

In the estimation of diffusivities for applications to chemical engineering problems, various approaches, kinetic theory, absolute theory, hydrodynamic theory, statistical-mechanical theory and both empirical and semi-empirical correlation had been employed for the calculation of binary diffusion coefficient. It is essential to appreciate that different theories are necessary for non-electrolytes and electrolytes solutions, therefore, different estimation methods are required for each case. A fact that all methods had been overlooked and limitations of each one were recognized too. 

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