## Mesoscopic non-equilibrium thermodynamics

It is clear from the discussion of the previous Section that issues of fluctuations and entropy, on shor t time and length scales, play an impor tant role in ultrafast biochemistry. Fur thermore, a biological system is of necessity in a state of non-equilibrium; “equilibrium” is a synonym for “death”. The question arises, is there a general formalism, on a level of sophistication above that of statistical mechanics, for treating a non-equilibrium state on the scale of macro-molecular fluctuations? Such a formalism, Mesoscale Non-Equilibrium Thermodynamics (MNET), is being developed for just this purpose [25]. Classical thermodynamics provides general rules for continuum, macroscopic systems in thermal equilibrium, in terms of a few parameters, such as the extensive variables internal energy, volume and mass, and intensive variables such as temperature, pressure and chemical potential. The dependence of the entropy on these variables is given by the Gibbs equation:TdS = dE + pdV – μdM

(equilibrium thermodynamics) In a non-equilibrium situation, gradients in the intensive variables act as forces, causing extensive fluxes. Such cases are treated in the framework of non-equilibrium thermodynamics (NET), in which the system is considered to consist of many subsystems, each of which is described, still in a continuum approximation, by equilibrium thermodynamic variables and each of which experiences a non-negative change in entropy. Consider the simple case of one-dimensional mass diffusion at constant energy and volume (dE = 0, dV = 0). In NET, the Gibbs equation for non-equilibrium thermodynamics may now incorporate a chemical potential with explicit and implicit spatial dependences, and hence the expression for the entropy production should read (see Infobox):

Under cer tain assumptions (see Infobox), one obtains the well-known diffusion equation for the density:

where is the diffusion constant, and L is the Onsager coefficient. Although NET allows the description of irreversible processes, the fact that it still is based on a continuum picture makes it inappropriate to treat systems at the molecular level, where strong fluctuations are important. Fur thermore, the defining relationships, such as the dependence of the fluxes on the forces, are linear. This makes it impossible for NET to describe, for example, activated processes in biochemistry. To extend NET to very shor t time and length scales (accessible with the SwissFEL), the statistical approach of Mesoscopic Non- Equilibrium Thermodynamics (MNET) is called for. The star ting point is again the dynamics of the entropy of a system, this time based on the Gibbs entropy postulate:

which relates the deviation of entropy from its equilibrium value with the probability density P of finding the system in a state with par ticular values for its nonequilibrated degrees of freedom γ. These degrees of freedom could include, e.g., a par ticle velocity, a spin orientation or the conformation of a biomolecule. The chemical potential is now µ(γ,t) = (γ,t) + kT lnP(γ,t), where is the activation energy barrier in an internal space coordinate at equilibrium, and the flux is again a linear function of the driving force

The NET expression for entropy production then leads to the Fokker-Planck equation

where the diffusion constant is now and L(γ,P) is again an Onsager coefficient, this time dependent on the thermodynamic state P and the mesoscopic coordinates γ. Here one has an explicit formalism to treat on equal footing an enthalpic and an entropic barrier. In a manner similar to the dissipation-fluctuation theorem (see Chapter I), MNET provides a method to “determine the dynamics of a system from its equilibrium proper ties”. Mesoscopic Non-Equilibrium Thermodynamics has application in the study of transport processes in confined systems, such as those in the biochemistry of the cell [26]. “Transport at the mesoscale is usually affected by the presence of forces of different nature: direct interactions between par ticles, hydrodynamic interactions mediated by the solvent and excluded volume effects. The presence of such diversity of forces has a direct implication in the form of the energy landscape, which may exhibit a great multiplicity of local minima separated by barriers. Transpor t at those scales presents two main characteristics: it is intrinsically non-linear, and it is influenced by the presence of fluctuations, external driving forces, and gradients. MNET can be used to infer the general kinetic equations of a system in the presence of potential barriers, which in turn can be used to obtain the expressions for the current of par ticles and the diffusion coefficient” [25]. The fields of application of MNET, namely molecular systems undergoing rapid fluctuations, match well those of the SwissFEL, and it is anticipated that the two could form a tight symbiotic relationship of theory and experiment.

### Non-Equilibrium Thermodynamics (NET)

To derive the diffusion equation in the NET formalism [25] (where, for example, the chemical potential μ and the particle density ρ may depend on the position x), one begins with the Gibbs equation for entropy change, assuming constant temperature, energy and volume:Introducing the mass flux J via the conservation law and per forming integration by par ts (with J vanishing at the system boundary), one obtains:

This expression establishes that and are “conjugate” quantities, which, according to Onsager, are related by a linear flux-force relation:

where L is the Onsager coefficient. Inser ting this expression into the conservation law yields the diffusion equation for the density:

where is the diffusion constant.