A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Asymptotic behavior of a hierarchical system of learning automata

Learning automata arranged in a two-level hierarchy are considered. The automata operate in a stationary random environment and update their action probabilities according to the linear-reward- -penalty algorithm at each level. Unlike some hierarchical systems previously proposed, no information transfer exists from one level to another, and yet the hierarchy possesses good convergence properties. Using weak-convergence concepts it is shown that for large time and small values of parameters in the algorithm, the evolution of the optimal path probability can be represented by a diffusion whose parameters can be computed explicitly.

A hierarchical system of learning automata that can learn die globally optimal path

Systems of learning automata have been studied by various researchers to evolve useful strategies for decision making under uncertainity. Considered in this paper are a class of hierarchical systems of learning automata where the system gets responses from its environment at each level of the hierarchy. A classification of such sequential learning tasks based on the complexity of the learning problem is presented. It is shown that none of the existing algorithms can perform in the most general type of hierarchical problem. An algorithm for learning the globally optimal path in this general setting is presented, and its convergence is established. This algorithm needs information transfer from the lower levels to the higher levels. Using the methodology of estimator algorithms, this model can be generalized to accommodate other kinds of hierarchical learning tasks.

Testing approximate theories of first-order phase transitions on the two-dimensional Potts model

The two-dimensional,q-state (q>4) Potts model is used as a testing ground for approximate theories of first-order phase transitions. In particular, the predictions of a theory analogous to the Ramakrishnan-Yussouff theory of freezing are compared with those of ordinary mean-field (Curie-Wiess) theory. It is found that the Curie-Weiss theory is a better approximation than the Ramakrishnan-Yussouff theory, even though the former neglects all fluctuations. It is shown that the Ramakrishnan-Yussouff theory overestimates the effects of fluctuations in this system. The reasons behind the failure of the Ramakrishnan-Yussouff approximation and the suitability of using the two-dimensional Potts model as a testing ground for these theories are discussed.

A Hierarchical System of Learning Automata

A learning automaton operating in a random environment updates its action probabilities on the basis of the reactions of the environment, so that asymptotically it chooses the optimal action. When the number of actions is large the automaton becomes slow because there are too many updatings to be made at each instant. A hierarchical system of such automata with assured c-optimality is suggested to overcome that problem.The learning algorithm for the hierarchical system turns out to be a simple modification of the absolutely expedient algorithm known in the literature. The parameters of the algorithm at each level in the hierarchy depend only on the parameters and the action probabilities of the previous level. It follows that to minimize the number of updatings per cycle each automaton in the hierarchy need have only two or three actions.

A cooperative game of a pair of learning automata

A cooperative game played in a sequential manner by a pair of learning automata is investigated in this paper. The automata operate in an unknown random environment which gives a common pay-off to the automata. Necessary and sufficient conditions on the functions in the reinforcement scheme are given for absolute monotonicity which enables the expected pay-off to be monotonically increasing in any arbitrary environment. As each participating automaton operates with no information regarding the other partner, the results of the paper are relevant to decentralized control.

Mean-field theories of Hubbard and t-J models

We report novel results obtained for the Hubbard and t-J models by various mean-field approximations.

Solvation dynamics in a Brownian dipolar lattice. Comparison between computer simulation and various molecular theories of solvation dynamics

Several recent theoretical and computer simulation studies have considered solvation dynamics in a Brownian dipolar lattice which provides a simple model solvent for which detailed calculations can be carried out. In this article a fully microscopic calculation of the solvation dynamics of an ion in a Brownian dipolar lattice is presented. The calculation is based on the non‐Markovian molecular hydrodynamic theory developed recently. The main assumption of the present calculation is that the two‐particle orientational correlation functions of the solid can be replaced by those of the liquid state. It is shown that such a calculation provides an excellent agreement with the computer simulation results. More importantly, the present calculations clearly demonstrate that the frequency‐dependent dielectric friction plays an important role in the long time decay of the solvation time correlation function. We also find that the present calculation provides somewhat better agreement than either the dynamic mean spherical approximation (DMSA) or the Fried–Mukamel theory which use the simulated frequency‐dependent dielectric function. It is found that the dissipative kernels used in the molecular hydrodynamic approach and in the Fried–Mukamel theory are vastly different, especially at short times. However, in spite of this disagreement, the two theories still lead to comparable results in good agreement with computer simulation, which suggests that even a semiquantitatively accurate dissipative kernel may be sufficient to obtain a reliable solvation time correlation function. A new wave vector and frequency‐dependent dissipative kernel (or memory function) is proposed which correctly goes over to the appropriate expressions in both the single particle and the collective limits. This form is expected to lead to better results than all the existing descriptions.

Ultrafast underdamped solvation: Agreement between computer simulation and various theories of solvation dynamics

A theoretical analysis of the three currently popular microscopic theories of solvation dynamics, namely, the dynamic mean spherical approximation (DMSA), the molecular hydrodynamic theory (MHT), and the memory function theory (MFT) is carried out. It is shown that in the underdamped limit of momentum relaxation, all three theories lead to nearly identical results when the translational motions of both the solute ion and the solvent molecules are neglected. In this limit, the theoretical prediction is in almost perfect agreement with the computer simulation results of solvation dynamics in the model Stockmayer liquid. However, the situation changes significantly in the presence of the translational motion of the solvent molecules. In this case, DMSA breaks down but the other two theories correctly predict the acceleration of solvation in agreement with the simulation results. We find that the translational motion of a light solute ion can play an important role in its own solvation. None of the existing theories describe this aspect. A generalization of the extended hydrodynamic theory is presented which, for the first time, includes the contribution of solute motion towards its own solvation dynamics. The extended theory gives excellent agreement with the simulations where solute motion is allowed. It is further shown that in the absence of translation, the memory function theory of Fried and Mukamel can be recovered from the hydrodynamic equations if the wave vector dependent dissipative kernel in the hydrodynamic description is replaced by its long wavelength value. We suggest a convenient memory kernel which is superior to the limiting forms used in earlier descriptions. We also present an alternate, quite general, statistical mechanical expression for the time dependent solvation energy of an ion. This expression has remarkable similarity with that for the translational dielectric friction on a moving ion.

Evaluating competing theories of street entrepreneurship

Conventionally, street entrepreneurs were either seen as a residue from a pre-modern era that is gradually disappearing (modernisation theory), or an endeavour into which marginalised populations are driven out of necessity in the absence of alternative ways of securing a livelihood (structuralist theory). In recent years, however, participa-tioninstreetentrepreneurshiphas beenre-read eitherasa rationaleconomicchoice(neo-liberal theory) or as conducted for cultural reasons (post-modern theory). The aim of this paper is to evaluate critically these competing explanations for participation in street entrepreneurship. To do this, face-to-face interviews were conducted with 871 street entrepreneurs in the Indian city of Bangalore during 2010 concerning their reasons for participation in street entrepreneurship. The finding is that no one explanation suffices. Some 12 % explain their participation in street entrepreneurship as necessity-driven, 15 % as traditional ancestral activity, 56 % as a rational economic choice and 17 % as pursued for social or lifestyle reasons. The outcome is a call to combine these previously rival explanations in order to develop a richer and more nuanced theorisation of the multifarious motives for street entrepreneurship in emerging market economies.

Spontaneous generation of optical activity in urea inclusion compounds vis-a-vis current theories

A novel test of recent theories of the origin of optical activity has been designed based on the inclusion of certain alkyl 2-methylhexanoates into urea channels.