Time-dependent thermodynamic systems

In [4] there is an analj is of time-dependent thermodynamic systems by obtaining the entropy production in terms of the relevant relaxation times in the system. The various physically possible limits of these relaxation times and their ratios leads to a classification into reversible, both quasistatic and otherwise, and irreversible processes. In one of these limits it is possible for a reversible process not to be quasistatic, but this limit is not physically interesting since for thermal conduction it would require infinite thermal conductivity. Hence our statement that power output requires irreversible processes is here substantiated. 

With slow up-pumping only a time-dependent thermodynamic temperature T(t) is needed to describe the state. In a nonequilibrium system where phonons and vibrations are pumped at different rates, a much larger number of parameters are needed to specify the state. Dlott and Fayer simplified... 

Promotion of a number of sub critical clusters to critical nuclei without growth, caused by reduction of critical cluster volume in time, is athermal nucleation. The concept of athermal nucleation, introduced by Fisher and Turnbull [43], consists in changing thermodynamic criterion of cluster stability. General expression of athermal nucleation in the systems with time-dependent thermodynamic parameters was derived in [21,45]. Angular distribution of athermal nucleation in the transient system is proportional to the distribution of critical clusters and the time derivative of the critical cluster volume... 

In noneqttilibrittm thermodynamics, one usually divides systems into two distinct types, namely, discrete composite systems and continuous systems. Discrete composite systems are systems for which the time-dependent thermodynamic properties are discontinuous across the boimdaries between the subsystems constimting the composite system. Continuous systems are systems for which thermodynamic properties vary continuously throughout. 

The coordinates of thermodynamics do not include time, ie, thermodynamics does not predict rates at which processes take place. It is concerned with equihbrium states and with the effects of temperature, pressure, and composition changes on such states. For example, the equiUbrium yield of a chemical reaction can be calculated for given T and P, but not the time required to approach the equihbrium state. It is however tme that the rate at which a system approaches equihbrium depends directly on its displacement from equihbrium. One can therefore imagine a limiting kind of process that occurs at an infinitesimal rate by virtue of never being displaced more than differentially from its equihbrium state. Such a process may be reversed in direction at any time by an infinitesimal change in external conditions, and is therefore said to be reversible. A system undergoing a reversible process traverses equihbrium states characterized by the thermodynamic coordinates. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

Since there is a flux even in the steady state, we need to know the flux of material and energy. Here, we hit very difficult problems but without these flows there is no life, and no development. Since there is flux of energy and material from and to the environment, the system is not open to description in thermodynamic variables alone - we require time-dependencies. 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

In a dynamic and cross-linkable system, such as the curing of a thermoset that contains a thermoplastic, the phase separation is more complicated than nonreaction system. The phase separation is controlled by the competing effects of thermodynamics and kinetics of phase separation and cure rate of thermoset resin (i.e. time dependent viscosity of the system). 

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

The need to reliably describe liquid systems for practical purposes as condensed matter with high mobility at a given finite temperature initiated attempts, therefore, to make use of statistical mechanical procedures in combination with molecular models taking into account structure and reactivity of all species present in a liquid and a solution, respectively. The two approaches to such a description, namely Monte Carlo (MC) simulations and molecular dynamics (MD), are still the basis for all common theoretical methods to deal with liquid systems. While MC simulations can provide mainly structural and thermodynamical data, MD simulations give also access to time-dependent processes, such as reaction dynamics and vibrational spectra, thus supplying — connected with a higher computational effort — much more insight into the properties of liquids and solutions. 

Kinetics deals with many-particle systems (thermodynamic ensembles). The properties measured as a function of time depend on the scale of observation, and this scale is chosen in relation to the question we wish to ask. The smaller the scale, the more inhomogeneous and fluctuating the homogeneous systems appear to be. For example, we describe the activated atomic jump frequency v as... 

Equation (5.1) described the vibrational response of a single particle to an applied forceF(t). In a (crystalline) system of many mobile particles (ensemble), the problem is analogous but the question now is how the whole system responds to an external force or perturbation Let us define the system s state (a) as a particular configuration of its particles and the probability of this state as pa. In a thermodynamic system, transitions from an a to a p configuration occur as thermally activated events. If the transition frequency a- /5 is copa and depends only on a and / (Markovian), the time evolution of the system is given by a master equation which links atomic and macroscopic parameters (dynamics and kinetics)... 

With electrochemical methods, we determine thermodynamic potentials of components in systems which contain a sufficiently large number of atomic particles. Since the systematic investigation of solid electrolytes in the early 1920 s, it is possible to change the mole number of a component in a crystal via the corresponding flux across an appropriate electrolyte (1 mA times 1 s corresponds to ca. 10 s mol). Simultaneously, the chemical potential of the component can be determined with the same set-tip under open circuit conditions. Provided both the response time and the buffer capacity of the galvanic cells are sufficiently small, we can then also register the time dependence of the component chemical potentials in the reacting solids. ... 

In an irreversible process, in conformity with the second law of thermodynamics, the magnitude that determines the time dependence of an isolated thermodynamic system is the entropy, S [23-26], Consequently, in a closed system, processes that merely lead to an increase in entropy are feasible. The necessary and sufficient condition for a stable state, in an isolated system, is that the entropy has attained its maximum value [26], Therefore, the most probable state is that in which the entropy is maximum. 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

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