Boltzmann equilibrium distribution thermodynamics

Over the past several years, there has been a renewed interest in thermodynamics and many scientists have considered it from new points of view [1-8]. Thermodynamics is a universal effective theory [9]. It does not depend on the details of underlying dynamics. The first law is the conservation of energy. The second law is the nonnegativeness of excess heat production. It is valid for wide classes of Markov processes in which systems approach to the Boltzmann equilibrium distribution. 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

In this subsection we will review the thermodynamics for the Boltzmann equilibrium distribution [7]. 

A fluid composed of a single species is described by five fields the three components of the velocity, the mass density, and the temperature. This is a drastic reduction of the full description in terms of all the degrees of freedom of the particles. This reduction is possible by assuming the local thermodynamic equilibrium according to which the particles of each fluid element have a Maxwell-Boltzmann velocity distribution with local temperature, velocity, and density. This local equilibrium is reached on time scales longer than the intercollisional time. On shorter time scales, the degrees of freedom other than the five fields manifest themselves and the reduction is no longer possible. 

Here the F) denote the positions of the monomers, so that r) specifies the conformation of the polymer chain. / ( /v, t F, 0) is the conditional probability density for a transition from r at time 0 to F at time t. The forces Fj are defined thermodynamically via the configurational equilibrium distribution function p( F ) at absolute temperature T (ks is Boltzmann s constant) ... 

Rudnicki and Niezgodka [43] used a model to simulate the melting of cocoa butter in DSC samples. They did not study crystallization from the melt, but they looked at how phases of cocoa butter transform from the most metastable to the most stable and finally to the liquid when the sample is heated. They made the following assumptions Transitions between phases are given by a bistable thermodynamic potential. The equilibrium distribution between the phases is described by Boltzmann functions, and process dynamics can be calculated by Arrhenius laws where energy levels of each phase are given by thermodynamics. 

The aim of this section is to give the steady-state probability distribution in phase space. This then provides a basis for nonequilibrium statistical mechanics, just as the Boltzmann distribution is the basis for equilibrium statistical mechanics. The connection with the preceding theory for nonequilibrium thermodynamics will also be given. 

The above derivation shows that Jarzynski s identity is an immediate consequence of the Feynman-Kac theorem. This connection has not only theoretical value, but is also useful in practice. First, it forms the basis for an equilibrium thermodynamic analysis of nonequilibrium pulling experiments [3, 15]. Second, it helps in deriving a Jarzynski identity for dynamics using thermostats. Moreover, this derivation clarifies an important aspect trajectories can be thought of as mapping initial conditions (I = 0) to trajectory endpoints, and the Boltzmann factor of the accumulated work reweights that map to give the desired Boltzmann distribution. Finally, it can be easily extended to transformations between steady states [17] in which non-Boltzmann distributions are stationary. 

Readout of the ligand information by a substrate is achieved at the rates with which L and S associate and dissociate it is thus determined by the complexation dynamics. In a mixture of ligands Li, L2. .. L and substrates Si, S2. . - S , information readout may assume a relaxation behaviour towards the thermodynamically most stable state of the system. At the absolute zero temperature this state would contain only complementary LiSi, L2S2. .. L S pairs at any higher temperature this optimum complementarity state (with zero readout errors) will be scrambled into an equilibrium Boltzmann distribution, containing the corresponding readout errors (LWS , n n ), by the noise due to thermal agitation. 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

Rahaman and Hatton [152] developed a thermodynamic model for the prediction of the sizes of the protein filled and unfilled RMs as a function of system parameters such as ionic strength, protein charge, and size, Wq and protein concentration for both phase transfer and injection techniques. The important assumptions considered include (i) reverse micellar population is bidisperse, (ii) charge distribution is uniform, (iii) electrostatic interactions within a micelle and between a protein and micellar interface are represented by nonlinear Poisson-Boltzmann equation, (iv) the equilibrium micellar radii are assumed to be those that minimize the system free energy, and (v) water transferred between the two phases is too small to change chemical potential. 

When a system is in thermodynamic equilibrium the level population, i.e. the number of atoms A in the excited state, is given by the Boltzmann distribution law ... 

Every one-, two- or three-dimensional crystal defect gives rise to a potential field in which the various lattice constituents (building elements) distribute themselves so that their thermodynamic potential is constant in space. From this equilibrium condition, it is possible to determine the concentration profiles, provided that the partial enthalpy and entropy quantities and jj(f) of the building units i are known. Let us consider a simple limiting case and assume that the potential field around an (planar) interface is symmetric as shown in Figure 10-15, and that the constituent i dissolves ideally in the adjacent lattices, that is, it obeys Boltzmann statistics. In this case we have... 

For spectra corresponding to transitions from excited levels, line intensities depend on the mode of production of the spectra, therefore, in such cases the general expressions for moments cannot be found. These moments become purely atomic quantities if the excited states of the electronic configuration considered are equally populated (level populations are proportional to their statistical weights). This is close to physical conditions in high temperature plasmas, in arcs and sparks, also when levels are populated by the cascade of elementary processes or even by one process obeying non-strict selection rules. The distribution of oscillator strengths is also excitation-independent. In all these cases spectral moments become purely atomic quantities. If, for local thermodynamic equilibrium, the Boltzmann factor can be expanded in a series of powers (AE/kT)n (this means the condition AE < kT), then the spectral moments are also expanded in a series of purely atomic moments. 

ORM assumes that the atmosphere is in local thermodynamic equilibrium this means that the temperature of the Boltzmann distribution is equal to the kinetic temperature and that the source function in Eq. (4) is equal to the Planck function at the local kinetic temperature. This LTE model is expected to be valid at the lower altitudes where kinetic collisions are frequent. In the stratosphere and mesosphere excitation mechanisms such as photochemical processes and solar pumping, combined with the lower collision relaxation rates make possible that many of the vibrational levels of atmospheric constituents responsible for infrared emissions have excitation temperatures which differ from the local kinetic temperature. It has been found [18] that many C02 bands are strongly affected by non-LTE. However, since the handling of Non-LTE would severely increase the retrieval computing time, it was decided to select only microwindows that are in thermodynamic equilibrium to avoid Non-LTE calculations in the forward model. 

All the work just mentioned is rather empirical and there is no general theory of chemical reactions under plasma conditions. The reason for this is, quite obviously, that the ordinary theoretical tools of the chemist, — chemical thermodynamics and Arrhenius-type kinetics - are only applicable to systems near thermodynamic and thermal equilibrium respectively. However, the plasma is far away from thermodynamic equilibrium, and the energy distribution is quite different from the Boltzmann distribution. As a consequence, the chemical reactions can be theoretically considered only as a multichannel transport process between various energy levels of educts and products with a nonequilibrium population20,21. Such a treatment is extremely complicated and - because of the lack of data on the rate constants of elementary processes — is only very rarely feasible at all. Recent calculations of discharge parameters of molecular gas lasers may be recalled as an illustration of the theoretical and the experimental labor required in such a treatment22,23. ... 

Entropy is also a macroscopic and statistical concept, but is extremely important in understanding chemical reactions. It is written in stone (literally it is the inscription on Boltzmann s tombstone) as the equation connecting thermodynamics and statistics. It quantifies the second law of thermodynamics, which really just asserts that systems try to maximize S. Equation 4.29 implies this is equivalent to saying that they maximize 2, hence systems at equilibrium satisfy the Boltzmann distribution. 

Because the density of charge is constant in each layer, the electric field varies linearly with distance between two boundaries, which implies, in turn, that the variation of the potential is quadratic within the same limits. Furthermore, we assume that charge transfer between adjacent layers is sufficiently efficient that the distribution of charge-carriers in the whole film is at thermodynamic equilibrium. Assuming Boltzmann s statistics holds, which is true as long as the charge density remains much lower than the density of molecules, this yields ... 

Although Boltzmann did not fully succeed in proving the tendency of the world to go to a final equilibrium state, there remain after all criticisms the following valuable results first, the derivation of the Maxwell-Boltzmann distribution for equilibrium states, then the kinetic interpretation of the entropy by the //-function, and finally the explanation of the existence of an integrating factor for dU+dA. In thermodynamics the existence of such a factor is always based on an unexplained hypothesis. 

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