Third day of Lux Out-of-Equilibrium. At the poster session yesterday night I learnt about a nice “distance from equilibrium” that characterizes nonequilibrium systems and that is connected to the entropy production rate of a system (T. Platini, PRE 2011). While the author employs a distribution of current/activities along a network, it might be feasible to generalize these results to the large time statistics of the joint current/activities and to discuss further examples, e.g. linear reaction networks.
– Gravitaxix of active Brownian particles and collective phenomena (C. Bechinger). E. Coli bacteria are able to move in preferential directions by means of long flagella with helicity, which render these organisms asymmetric. This is the kind of behavior known as active Brownian motion of small microsimmers. Such particles are structured in such a way that the environment acts in an asymmetric manner on them. An example are particles consisting of two hemispheres of different metals (platinum-gold?), one of which is responsible of the oxidation (?) of the hydrogen peroxide solvent. The question has arisen whether such systems could just be described as normal colloidal system with an effective temperature, but this is not the case: an initially homogeneous equilibrium distribution of E.Coli evolves into an inhomogeneous steady state, and again such an ensemble of E.Coli can move a ratched in a preferred direction, hence violating equilibrium. Bechinger reports on experimental work mostly, analyzing problems such as microswimmers in a crowd, phase separation of self-propelled particles, etc.
– Tipicality of eigenstate thermalization (P. Reimann): One recurrent question in thermodynamics is how irreversibility emerges from underlying reversible laws of nature. In the context of Quantum Mechanics, the underlying evolution is dictated by unitary dynamics and a proposal is that if the so-called Eigenstate Thermalization Hypothesis holds, then all looks as if a microcanonical description is valid. The ETH states that if energies are close, then the value of an observable in the energy eigenstate are also close. In this sense thermal equilibrium is “encapsulated” in every eigenstate of the Hamiltonian, and this fact is initially hidden by a “conspiracy” of the off-diagonal elements (the coherences, which by recurrence are sooner or later to reappear and screw everything up again). This field was initiated by Deutsch, Srednicki, Rigol et al. in the early 90’s. Counterexamples to the ETH are easy to build, so the question is whether it holds for “realistic” Hamiltonians, e.g. sampled randomly from a random matrix ensemble (Deutsch 1991), the assumption holds. But here I missed a point: is ETH a property of the Hamiltonian or of a set of observables, or of any observable given a certain Hamiltonian? Also, I must say I am not very enthusiastic about the very premises of this kind of research, in that from a philosophical point of view I believe that thermodynamics applies to intrinsically open systems and that the question whether it could emerge in an isolated system does not make perfect sense (I will come back to this at some point). In fact, it is interesting to note that in any such tentative explanations of thermodynamics from dynamics people always implicitly introduce some form of dissipation or loss of information, e.g. the randomness in Deutsch’s argument which plays the role of the tipicality of Boltzmann against Zermelo’s argument that a recurrent dynamical system must always return the entropy to its initial value.