Objectives
For the last 40 years, the interplay between simulation and analytical theory has provided a new perspective for such basic notions as temperature, fluctuations, thermostats and non-equilibrium ensembles, and even for the Second Law [VdB2010]. Since these results are of ever-increasing importance for nano-scale physics, chemistry, materials science, and biology, the present workshop aims at assessing the present state and most promising developments in the future. Below we identify only a few issues which are open problems and which will be discussed at the workshop. Of course, this list is incomplete and will be augmented by the participants.
Can Boltzmann's and Gibbs' ensemble theory be extended to stationary nonequilibrium states? In equilibrium, the free energy and the entropy are well-defined objects, which determine the equilibrium distribution and, hence, the macroscopic properties of a systems as well as its fluctuations. No generalization of these functions have been found for nonequilibrium systems far from equilbrium. For a class of dynamically-thermostatted models these functions even diverge. However, the global rate of entropy production is well defined and remains finite. The question arises, whether for infinite systems a local rate of entropy production may be defined [R2008].
As stated recently by J.L. Lebowitz in a list of important open problems [L2010], the constitutive relations, used to close the set of hydrodynamic equations for actual computations, still lack a derivation from microscopic dynamics.
To achieve a nonequilibrium stationary state, thermostats are required, which remove the irreversibly generated heat [E2010]. Recently, the efficiency of such methods has been re-examined [L2010]. Various alternatives to traditional methods such as Langevin dynamics appeared, which introduce stochastic perturbations in novel ways and offer greater flexibility. This includes the stochastic scaling methods and the Nose-Hoover-Langevin method [S2007].
For systems, for which the ‘chaotic hypothesis’ does not hold, such as non-uniformly hyperbolic systems, new phenomena may arise. First, the nonequilibrium stationary state (NESS) may depend non differentially on the parameters. Whether this is experimentally observable remains to be seen. Second, the dispersion relations of linear response theory may be violated such that the susceptibilities have singularities in the upper half of the complex plane. This may cause a ‘passive’ system to become ‘active’: if a periodic external perturbation is applied, the system transfers energy to the outside world and becomes an amplifying device, if the active state is maintained [R2009]. An ‘active’ NESS strongly differs from an equilibrium state. It is an open question, what happens with the stationary fluctuation theorem in such a case.
Work in dynamical systems theory has provided new mathematical tools, such as the Sinai-Ruelle-Bowen measures. The application to hyperbolic systems has led to the derivation of fluctuation theorems, the only exact relations for far-from-equilibrium systems. Another important current topic has been the development of work theorems for the computation of free-energy differences with non-equilibrium means. These theorems relate equilibrium free energies to the statistics of work carried out while driving the system away from equilibrium and are valid under very general conditions for deterministic as well as stochastic dynamics. The generalization of these relations and their application to various systems in areas ranging from materials science to biology will also be a topic at the proposed workshop, as will be the discussion of recent results for heat conduction in low-dimensional systems and the study of non-extensive systems with negative heat capacity.
Rigorous results have been obtained for exclusion processes and related models. They include fluctuation theorems, scaling properties of correlation functions and a characterization of hydrodynamic equations. This field is a testing ground and source of inspiration for more general but less rigorous theories.
The present conference will focus on the most recent developments in the theory of nonlinear dynamics and their mutual relationships to stochastic dynamics as well as its applications to fields ranging from physic and chemistry to materials science, non-linear spectroscopy and molecular biology. It is hoped that the meeting will enhance communication in the community and will lead to the development of new ideas and computational and theoretical approaches.
[E2010] D.J. Evans, D.J. Searles, and S.R. Williams, Musings on thermostats, J. Chem. Phys. 133, 104106 (2010).
[L2010] J.L. Lebowitz, Round table disussion at the 11th Granada Seminar, Foundations of Nonequilibrium Statistical Physics, La Herradura, 2010.
[L2010] B. Leimkuhler, E. Noorizadeh and O. Penrose, On the efficiency of temperature controls for molecular dynamics, preprint (2010).
[R2008] D. Ruelle, What physical quantities make sense in nonequilibrium statistical mechanics, in Boltzmann’s Legacy edited by G. Gallavotti, W.L. Reiter, and J. Yngvason, European Mathematical Society, Zürich, 2008.
[R2009] D. Ruelle, A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855 (2009).
[S2007] A. Samoletov, C. Dettmann, and M Chaplain, Thermostats for “slow” configurational modes, J. Stat. Phys. 128, 1321 (2007).
[VdB2010] C. Van den Broeck, J. Stat. Mech: Theory and Experiment, "The many faces of the second law", P10009 (2010).