Thermodynamic laws, fluctuation-dissipation

Molecular dynamics simulations entail integrating Newton s second law of motion for an ensemble of atoms in order to derive the thermodynamic and transport properties of the ensemble. The two most common approaches to predict thermal conductivities by means of molecular dynamics include the direct and the Green-Kubo methods. The direct method is a non-equilibrium molecular dynamics approach that simulates the experimental setup by imposing a temperature gradient across the simulation cell. The Green-Kubo method is an equilibrium molecular dynamics approach, in which the thermal conductivity is obtained from the heat current fluctuations by means of the fluctuation-dissipation theorem. Comparisons of both methods show that results obtained by either method are consistent with each other [55]. Studies have shown that molecular dynamics can predict the thermal conductivity of crystalline materials [24, 55-60], superlattices [10-12], silicon nanowires [7] and amorphous materials [61, 62]. Recently, non-equilibrium molecular dynamics was used to study the thermal conductivity of argon thin films, using a pair-wise Lennard-Jones interatomic potential [56]. 

FIGURE 1 Distribution of work values for many repetitions (realizations) of a thermodynamic process involving a nanoscale system. The process might involve the stretching of a single RNA molecule or, perhaps, the compression of a tiny quantity of gas by a microscopic piston. The tail to the left of AF represents apparent violations of the second law of thermodynamics. The distribution p( W) satisfies the nonequilibrium work theorem (Eq.6), which reduces to the fluctuation-dissipation theorem (Eq.l) in the special case of near-equUibrium processes. 

Before we come to these models, we will first introduce a basic law of statistical thermodynamics which we require for the subsequent treatments and this is the fluctuation-dissipation theorem . We learned in the previous chapter that the relaxation times showing up in time- or frequency dependent response functions equal certain characteristic times of the molecular dynamics in thermal equilibrium. This is true in the range of linear responses, where interactions with applied fields are always weak compared to the internal interaction potentials and therefore leave the times of motion unchanged. The fluctuation-dissipation theorem concerns this situation and describes explicitly the relation between the microscopic dynamics in thermal equilibration and macroscopic response functions. 

Thermodynamics had been studied both in far-from-equilibrium and in near-equilibrium situations. A near-equilibrium world is a stable world. Fluctuations regress. The system returns to equilibrium. The situation changes dramatically far from equilibrium. Here fluctuations may be amplified. As a result, new space-time structures arise at bifurcation points. We considered the possibility of oscillating reactions as early as in 1954, many years before they were studied systematically. We introduced concepts such as selforganization and dissipative structures, which became very popular. In short, irreversible processes associated to the flow of time have an important constructive role. Therefore, the question that arises is how to incorporate the direction of time into the fundamental laws of physics, be they classical or quantum. 

Nevertheless, this self-organization does not contradict the second law of thermodynamics because the total entropy of the open system keeps increasing, but this increase is not uniform throughout disorder. In fact, such dissipative structures are the islands (fluctuations) of order in the sea (background) of disorder, maintaining (and even increasing) their order at the expanse of greater disorder of their environment. 

According to the second law of thermodynamics, ary spontaneous process in an isolated system out of equilibrium will lead to an increase in the errtropy inside that system. In spite of the general validity of the second law, we have yet to fully rmderstand the problem of irreversible time evolution. Of course. Brownian motion theory does provide some insight into the direction that might be pursued in seeking answers to this problerrr. The most impor-tarrt idea to emerge from Brownian motion theory is the notion that dissipative or irreversible behavior arises from spontaneous equilibrium fluctuations. 

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