Chap 2
Opening Chapter 2
Describing linear response theory at a basic level is quite a challenging task, because it contains a great deal of physical hypotheses and mathematical difficulties. The former have to be carefully justified by heuristic arguments, while the latter have to be illustrated in detail to allow for a clear understanding of most of the physical consequences. On top of that, there is the additional difficulty of casting the main concepts into a frame, where the unexperienced reader can recognize a logical and pedagogical organization. This is why we begin this chapter showing how we can obtain the Kubo relation for a Brownian particle. Our aim is to point out from the very beginning the importance of equilibrium time correlation functions as basic ingredientsfor an approach to nonequilibrium phenomena. In order to explore the possibility of extending a similar approach to nonequilibrium thermodynamic quantities we have to recognize that, on a formal ground, a thermodynamic observable can be assumed to be a fluctuating quantity, even at equilibrium (the amplitude of its relative fluctuations vanishing in the thermodnamic limit). In practice, a preliminary step in the direction of a linear response theory of nonequilibrium processes amounts to assume that thermodyamic observables obey a fluctuation-driven evolution, described bygeneralized Langevin and Fokker-Planck equations, in full analogy with a Brownian particle. Despite a rigorous mathematical formulation should be in order, here we justify this assumption on the basis of physical plausibility arguments.
An explicit dynamical description of the physical mechanism of fluctuations of a thermodynamic observable, as the result of the presence of an associated perturbation field switched on in the past, allows to establish a quantitative relation with the way this thermodynamic observable relaxes to its new equilibrium value, when the perturbation field is switched off. In particular, the relaxation mechanism is found to be ruled by the decay rate of the equilibrium time correlation function of the perturbed thermodynamic observable.
Now we can come back to transport processes relying upon effective theoretical tools. In fact, generalized Green-Kubo relations for the transport coefficients like diffusivity, viscosity and thermal conductivity of a general thermodynamic system can be expressed in terms of the equilibrium time correlation functions of the mass, momentum and energy currents, respectively. These results are obtained by exploiting the formal equivalence of the Fokker-Planck equation with a continuity equation, thus providing a hydrodynamic basis to transport phenomena. The Onsager regression relation allows to extend such a description to coupled transport processes.
The bridge between linear response theory and the formulation of the thermodynamics of irreversible processes can be established byconsidering entropy as a genuine dynamical variable: its production rate, that can be expressed as a combination of generalized thermodynamic forces (affinities) and the corresponding fluxes, is the key ingredient for characterizing transient as well as stationary nonequilibrium conditions. Combining this result with the Onsager reciprocity relations and the Onsager theorem for charged transport we can formulate a phenomenological theory of coupled transport processes, which is summarized by the Onsager matrix of generalized transport coefficients: its properties and symmetries again originate from those of equilibrium time correlation functions of pairs of thermodynamic observables. We point out that, despite entropy is treated as a dynamical variable, we assume that in a linear regime it maintains the same dependence of the equilibrium case with all the basic thermodynamic quantities, like internal energy, pressure and chemical potentials. This amounts to assert that also for stationary nonequilibrium processes associated to coupled transport phenomena, local equilibrium conditions hold.
The final part of this chapter is devoted to a short illustration of linear response theory for quantum systems. Despite the formulation is not particularly involved, its application to specific examples requires the knowledge of Quantum Field Theory. For this reason, the last Sec. 2.10.2 can be easily understood only by the reader familiar with such topic.
Bibliography Chapter 2
- A classical textbook about linear response theory, fluctuation-dissipation theorem and their applications to nonequilibrium statistical mechanics is R. Kubo, M. Toda and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, 3rd edn (Springer, 1998). In particular, this book contains a rigorous extension of the formalism of stochastic differential equations to macroscopic observables by the so-called projection formalism.
- A classical textbook on nonequilibrium thermodynamics, containing a detailed presentation of the Onsager theory of transport processes is S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, 1984).
- The derivation of the hydrodynamic equations from the Boltzmann model of the ideal gas as a consequence of the conservation of mass, momentum and energy densities is contained in the book by K. Huang, Statistical Mechanics , 2nd edn (Wiley, 1987).
- A pedagogical introduction to entropy is the book by D. S. Lemons, A Student’s Guide to Entropy (Cambridge University Press, 2013).
- An advanced texbook on nonequilibrium statistical physics with applications to condensed matter problems is G. F. Mazenko, Nonequilibrium Statistical Mechanics (Wiley-VCH, 2006).
- Many applications of nonequilibrium many-body theory to condensed matter physics are contained in the book by P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 2000).
- An interesting and modern book about nonequilibrium statistical mechanics of turbulence, transport processes and reaction-diffusion processes is J. Cardy, G. Falkovich and K. Gawedzki, Non-Equilibrium Statistical Mechanics and Turbulence (Cambridge University Press, 2008). This book contains the lectures given by the authors at a summer school. This notwithstanding, these lectures concern frontier topics in nonequlibrium statistical mechanics that may not be immediately accessible to undergraduate students.
- A comprehensive and up-to-date account on anomalous transport phenomena in low-dimensional systems can be found in the book by S. Lepri (ed.) Thermal Transport in Low Dimensions, Lecture Notes in Physics (Springer, 2016). Some of the contributions contained in this book illustrate also experimental results.
- The peculiar thermodynamic properties of equilibrium and out of eqilibrium systems with long–range interactions are discussed in the book by A. Campa, T. Dauxois, D. Fanelli and S. Ruffo, Physics of Long-Range Interacting Systems (Oxford University Press, 2014).
- A mathematical textbook about analytic functions and their Fourier and Laplace transform, together with functional analysis is W. Rudin, Functional Analysis (McGraw-Hill, 1991).
- Many interesting aspects concerning the applications of linear response theory and of the Kubo formalism to linear irreversible thermodynamics are contained in the book by D. J. Evans and G. P. Morris, Statistical Mechanics of Nonequilibrium Liquids, 2nd edn (ANU E Press, 2007). This book contains also an instructive part, concerning the use of computer algorithms useful for performing numerical simulations of fluids.
- A short survey about models of thermal reservoirs useful for numerical simulations are discussed in Sec. 3 of the review paper by S. Lepri, R. Livi and A. Politi, Thermal Conduction in Classical Low-Dimensional Lattices (arXiv), Physics Reports, 377 (2003) 1-80.