State Of Maximum Entropy Research Articles

Entropy and chemical change. III. The maximal entropy (subject to constraints) procedure as a dynamical theory

An equivalence between the dynamical (equations of motion) and the information theoretic (maximal entropy) approaches to collision phenomena is established. The connection is demonstrated in both directions. The variational procedure of maximal entropy is shown to converge to an exact solution of the equations of motion (be they classical or quantal) throughout the collision. In particular, a stationary precollision state is proved to be a state of maximal entropy (subject to constants of the unperturbed motion) and to remain a state of maximal entropy throughout the collision. Conversely, the exact solution of the equations of motion is shown to be of maximal entropy. In this fashion one obtains an algebraic procedure for the specification of the constraints which determine (via the procedure of maximal entropy) an exact solution of the equations of motion. Surprisal analysis does not require the solution of differential equations. These must be solved to determine the magnitude of the Lagrange parameters (i.e., for surprisal synthesis). The number of coupled differential equations that have to be solved (to obtain an exact solution) equals the number of constraints. Sum rules (which express the mean value of any constraint as a linear function of the initial mean values of the constraints) are derived and offer an alternative route to surprisal synthesis. The information theoretic result for the branching ratio (as the ratio of two partition functions) is shown to be a rigorous result of the present formalism. As an illustration, a simple Hamiltonian for a collinear reactive collision is analytically treated in detail (for a classical motion along the reaction coordinate). The constraints are identified (with special reference to reactive collisions, where the Hamiltonians for the reactants and products do differ); the time dependence of the Lagrange parameters is established and the vibrational state distribution (both during and after the collision) is determined. A sum rule for the mean products vibrational energy is discussed. For a stationary initial state (e.g., a particular vibrational state or a thermal distribution), 〈E′vib〉 is linearly dependent on 〈Evib〉 alone. The slope and intercept are determined only by the dynamics and are essentially proportional to the ratio of the final to initial vibrational frequencies. For a family of related reactions where this ratio is nearly unchanged, the vibrational energy disposal would be quite similar. Inefficient products vibrational excitation is expected when this ratio is low.

Entropy production in the functional random-walk model

An information-theoretical background is presented here for the functional random-walk model of a many-particle system, which was recently proposed to simulate nonequilibrium statistical mechanics in a certain coarse-graining sense. Next, entropy productivity and the maximum-entropy state in the model dynamics are studied with the new definition of entropy, which turns out to be a natural extension of the original Boltzmann entropy.

Statistical thermodynamics of collisionless plasma. II. Stability theory of inhomogeneous systems

For pt. I see ibid., vol. 14, no. 5, 427. The statistical model developed in Part I is applied to the discussion of the stability of inhomogeneous Vlasov equilibria. The existence of instability is expressed by the general condition integral I(x)( theta (x)- phi 0)dV>0 provided a certain class of variations compatible with the physical constraints imposed on the system does exist. In this criterion I(x) is the information variable depending on the physical system under consideration, phi (x) the corresponding potential, and phi 0 its spatial average. When I(x) is the charge density the criterion can be applied in order to show the instability of the Bernstein, Greene and Kruskal solutions. It is also shown, by means of the entropy principle, that the isolated inhomogeneous collisionless systems evolve towards more homogeneous equilibria, associated with a higher entropy, by transforming their collective energy into the energy of the random individual fluctuations. Moreover, nearly homogeneous but electrostatically unstable collisionless plasmas prove to evolve towards the marginal state of the stable equilibrium (which is a state of maximum entropy for the collisionless systems), while an electrostatic quasi-static marginally unstable mode with long wavelength appears in the system.

Statistical characterization of polycrystalline ferrite media

The classical field theory superposition principle, the central limit theorem, and the gyromagnetic equations of motion are invoked as a basis to formulate a quasi-static statistical model for polycrystalline ferrite media with Gaussian statistics. The use of this model to analyze small signal reversible rotational phenomena is illustrated. It is also demonstrated that this model and a previous microwave model given by the authors are maximum entropy models An analogy between the maximum entropy property of these models and the Gibbs thermodynamic equivalence for equilibrium and a state of maximum entropy is noted.

Statistical Characterization of Polycrystalline ferrite media

The classical field theory superposition principle, the central limit theorem, and the gyromagnetic equations of motion are invoked as a basis to formulate a quasi-static statistical model for polycrystalline ferrite media with Gaussian statistics. The use of this model to analyze small signal reversible rotational phenomena is illustrated. It is also demonstrated that this model and a previous microwave model given by the authors are maximum entropy models An analogy between the maximum entropy property of these models and the Gibbs thermodynamic equivalence for equilibrium and a state of maximum entropy is noted.

Chemical equilibrium as a state of maximal entropy

The non-trivial disadvantages of the conventional approach to chemical equilibrium has prompted this author to reconsider the possibility of a direct exploitation of the simple entropy criterion in the discussion of chemical equilibrium.

Statistical properties of boolean variables generated by a class of binary arithmetic processes

The statistical properties of variables generated by some common binary arithmetic operations are discussed. The binary arithmetic considered are add, subtract, two’s complement transformation, multiply, and divide. It is demonstrated that the serially generated variables for all these operations are stochastic for all conditions except when the input variables are at a maximum or minimum entropy condition. For stationary inputs, serially generated variables generally tend to become non-stationary in the wide sense. The use of Markov chain to model the add and subtract processes is also presented.

Dynamics of Flattening in Rotating Stellar Systems

view Abstract Citations (3) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Dynamics of Flattening in Rotating Stellar Systems Milder, D. Michael Abstract A discussion of galactic dynamics involving elementary moments of the matter distribution verifies that orbital energy loss causes a system containing angular momentum to become flat and to approach solid-body rotation. The results confirm a generally accepted qualitative explanation of flattening in rotating systems given by Poincare. In addition, the discussion makes apparent that significant flattening during evolution must be accompanied by a several4old decrease in a galaxy's over-all size. Maximum entropy states are derived for gravitating systems and are shown to conform to the simple dynamical description used. Publication: The Astrophysical Journal Pub Date: July 1966 DOI: 10.1086/148745 Bibcode: 1966ApJ...145..109M full text sources ADS |

Conditions for Equilibrium at Negative Absolute Temperatures

In the classical phenomenological thermodynamic theory of Gibbs, equilibrium is defined as the state of maximum entropy at constant energy, and a theorem is proved (the energy theorem) which asserts that the equilibrium state is the state of minimum energy at fixed entropy. This theorem is not true for systems at negative absolute temperatures. By examining a familiar statistical model which can exhibit negative absolute temperatures we find the correct form of the energy theorem. It turns out that at negative absolute temperatures the state of equilibrium of a system with a given entropy is that in which the system has its highest energy.

A statistical mechanics of interacting biological species

The system of differential equations proposed by V. Volterra, desoribing the variation in time of the populationsNr of interacting species in a biological association, admits a Liouville's theorem (when logNr are used as variables) and a universal integral of “motion”. Gibbs' microcanonical and canonical ensembles can then provide a thermodynamic description of the association in the large. The “temperature” measures in one number common to all species the mean-square deviations of theNr from their average values. There are several equipartition theorems, susceptible of direct experimental test, a theorem on the flow of “heat” (the conserved quantity in an isolated association) between two weakly coupled associations at different temperatures, a Dulong-Petit law for the heat capacity, and an analog of the second law of thermodynamics expressing the tendency of an association to decline into an equilibrium state of maximal entropy. The analog of the Maxwell-Boltzmann law is a distribution of intrinsic abundance for each species which has been successfully used by ecologists for interpreting experimental data. A true thermodynamics develops upon introducing the idea of work done on an association through a variation of the variables (such as physical temperature) defining the physical and chemical environment. An ergodic theorem is suggested by the agreement of ensemble and time averages in the one case where the latter may be found explicitly.

Interaction between Rubber and Liquids. IV. Factors Governing the Absorption of Oil by Rubber

Abstract A piece of vulcanized rubber, dropped into benzene, swells to several times its size but retains its shape. With raw rubber, a further stage ensues, in which the rubber flows and ultimately disperses. Inevitably, one tends to form a picture of the rubber attracting and holding the liquid with some strong force. It is the purpose of this paper to explain why this picture is believed to be entirely false, and to give an alternative explanation of the phenomena of swelling and solution. It will be necessary first to consider briefly the way in which simpler materials mix with one another. The simplest possible system is that of two gases which do not react with each other. In a gas the molecules spend most of their time a long way from one another, and the total energy of the system is therefore made up largely of the kinetic energy of thermal motion. As a consequence of this kinetic energy the gas molecules tend, on the average, to distribute themselves uniformly, so that any pair of gases mix completely. A quantitative interpretation can be given to this mixing tendency, in terms of the concept of entropy. Of the various ways in which entropy may be regarded, the most useful for the present purpose is in terms of probability. Qualitatively it is evident that gas molecules in violent thermal agitation are extremely unlikely to arrange themselves so all molecules of one type are confined to one part of the vessel. This is expressed in thermodynamic language by saying that the entropy of such an arrangement would be small. The second law of thermodynamics states that if, as in a gas, there is no change of energy, a system tends to take up the state of maximum entropy, or maximum randomness. A quantitative expression to the relationship between the entropy S and probability W, takes the form:

Possibilities in Relativistic Thermodynamics for Irreversible Processes Without Exhaustion of Free Energy

Previous applications of relativistic thermodynamics have been either to systems in static thermodynamic equilibrium or to non-static systems in which reversible processes are taking place. The present article deals with non-static systems in which irreversible processes take place. It is shown that the failure of the principle of energy conservation to hold in relativistic mechanics in its older simple form removes a classical limitation on the irreversible increases in entropy which can take place in an isolated system. The removal of this limitation provides possibilities in relativistic thermodynamics for irreversible processes to take place in an isolated system without ever reaching an unsurpassable state of maximum entropy and minimum free energy where further change would be impossible. Such possibilities are found to be illustrated by non-static models of the universe which could undergo a continued series of irreversible expansions and contractions without ever arriving at a state of rest. Special attention is given to a non-static model of the universe in which the irreversible annihilation of matter would take place in the later stages of expansion and the irreversible formation of matter out of radiation in the later stages of contraction. Finally, remarks are made concerning the bearing of the findings on the interpretation of phenomena in the actual universe.

The Universe and Irreversibility

ON the assumptions that space-time is finite spatially but not temporally (apart from supernatural events such as creation), and that atoms and radiation are mutually convertible, Sir James Jeans (NATURE, 122, p. 689; 1928) arrives at the conclusion that the universe is progressing towards a final state of maximum entropy from which no return is possible. While such a state has a maximum a priori probability, it does not follow that it is final.