Boltzmann equilibrium distribution production

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. 

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell-Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity K(fhl, ij) may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium. 

It was suggested earlier that if the half life of a reaction was to be less than one millisecond, the product of the Boltzmann factor and the partial pressure of the species involved had to be greater than 10 . The consideration of time scales indicated that a reaction could only be considered to be balanced, that is, to depart negligibly from the equilibrium distribution of concentrations, if the half life was less than... 

The Maxwell-Boltzmann velocity distribution of the three-dimensional molecular velocity vector for a system at equilibrium with zero fluid velocity is equal to the product of each of the three independent normal distributions of the three independent velocity components ... 

Transition State Theory [1,4] is the most frequently used theory to calculate rate constants for reactions in the gas phase. The two most basic assumptions of this theory are the separation of the electronic and nuclear motions (stemming from the Bom-Oppenheimer approximation [5]), and that the reactant internal states are in thermal equilibrium with each other (that is, the reactant molecules are distributed among their states in accordance with the Maxwell-Boltzmann distribution). In addition, the fundamental hypothesis [6] of the Transition State Theory is that the net rate of forward reaction at equilibrium is given by the flux of trajectories across a suitable phase space surface (rather a hypersurface) in the product direction. This surface divides reactants from products and it is called the dividing surface. Wigner [6] showed long time ago that for reactants in thermal equilibrium, the Transition State expression gives the exact... 

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. 

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. ... 

So the equilibrium constant is a simple function of the difference in energy and the difference in number of available states between the reactants and the products. We can also understand Equation 4.47 by forgetting about the intermediate state, and just applying the Boltzmann distribution directly to the reactants and products. 

Bair and Fitzsimmons compared the observed concentration in v = 18 with its equilibrium concentration at 9800°K, which would be the vibrational temperature (again assuming a Boltzmann distribution) if the reaction energy was shared equally between the degrees of freedom of the products. However, their figure for (Nv=18lN)T=m0 j( should read 5.3 x 10 3 and not 5.3 x 10 4. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

Traditional literature treats enzyme catalyzed reactions, including hydrogen transfer, in terms of transition state theory (TST) [4, 34, 70]. TST assumes that the reaction coordinate may be described by a free energy minimum (the reactant well) and a free energy maximum that is the saddle point leading to product. The distribution of states between the ground state (GS, at the minimum) and the transition state (TS, at the top of the barrier) is assumed to be an equilibrium process that follows the Boltzmann distribution. Consequently, the reaction s rate is exponentially dependent on the reciprocal absolute temperature (1/T) as reflected by the Arrhenius equation ... 

In all the experiments mentioned above, the Overhauser effect has been observed by irradiating the e.s.r. signal of the dissolved free radicals. However the essential conditions for production of an Overhauser effect are that the populations of the electron spin Zeeman levels should depart from their thermal equilibrium value and that, as the electron spins relax and attempt to restore the Boltzmann distribution among their levels, they should interact with the nuclear spins present in solution. 

It is well known that rapid exothermic bimolecular elementary reactions can lead to product molecules in which the initial rotational and vibrational energy distributions are dissimilar to those of equilibrium at laboratory temperatures, Tq. A nonrigorous but useful description of such distributions is that the rotational and vibrational temperatures , and Ty (obtained by fitting the observed distributions to Boltzmann distributions for and Ty) are such that Tr> To <. Ty. It is possible for complete vibrational population inversion to occur in the initial products of reactions. This corresponds to the case when the population of an excited vibrational level 6(v) exceeds that of the vibrational ground state, 6(v) > 6(0), and leads to the unrealistic description Ty < 0. An intermediate case— partial inversion— is also observed when ly > 0 as rotational relaxation is more rapid than vibrational relaxation, Tq, and thus inversion exists over a limited range of rotational quantum numbers in respect of P(J) or R(J) transitions to the vibrational ground state. Laser action... 

By scanning the probe laser over one or more rotational branches of the product, the relative intensities of the lines in this excitation spectrum may be used to determine product rotational (and/or vibrational) state distributions. In order to arrive at fully quantitative answers, corrections have to be made for relative transition probabilities, fluorescence lifetimes of the excited state, and any wavelength-dependent detection functions (such as the detection system spectral response). But once this has been done, one can deduce the ground state distribution function(s) by examining the so-called excitation spectrum of a molecular species. For thermal equilibrium conditions, the level population /V, can be described using a Boltzmann distribution function with temperature as the most important parameter in its most general form this is... 

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