Entropy of irreversible cooling and the “discrimination” of model stochastic processes

Entropy changes are calculated for the irreversible cooling of a homogeneousN-particle system. The execution of an appropriate model stochastic process enables one to calculate the “discrimination”D (from the transition probabilities of the actual steps) and < ? D> is shown to be equal to the external entropy change ΔS _ext. This is trivially true for the “Metropolislike” processes, where the individual particles maintain a direct heat exchange with a reservoir. “Cooperative” processes, which attribute the heat exchange to the mass ofN particlesin toto, are also considered; for these ΔS _ext is still equal to < ? D>. Hence, knowing and the entropy of the initial and final states of the system, one can calculate the net entropy production and study its minimization. Alternatively, a consistently probabilistic approach (independent of thermodynamic equivalents) postulates that statistical mechanical processes proceed with the least discrimination, Min, for given conditions. The postulate is supported by its conformance with the second law of thermodynamics. Min reduces to the Jaynes principle both at equilibrium and for isolated systems. Computer experiments illustrating the calculation ofD are presented. These describe the cooling of a square Ising lattice, with the help of the Metropolis and of the cooperative model processes; the latter, optimized for least entropy production, rapidly converge toward equilibrium.     

Entropy and stability in a stochastic laser model

The asymptotic stability of a microscopic stochastic laser model in the Heisenberg picture is discussed using Liapounov's direct method. By reducing the laser Langevin equations to a classical van der Pol equation for the photon number in the rotating wave approximation, it may be shown that the above-threshold steady state characterized by the cubic non-linearity in the van der Pol equation, is stable. This leads to the definition of a monotonic increasing entropy function far from thermal equilibrium.     

Bolometers operated at 0.1 K and 0.2 K cooled by adiabatic demagnetization

We have constructed bolometers operating at temperatures of 0.1 K and 0.2 K cooled by adiabatic demagnetization. The electrical noise equivalent power (NEP) of the bolometers was found to be 1.8×10^–16 W/Hz^1/2 and 6.6×10^–16 W/Hz^1/2 respectively. These values are within a factor of two of the expected performance based upon bolometer theory. The thermistor material used was gallium doped single crystal germanium, for which we find a resistivity temperature dependence of the form = ^o exp (AT^–1/2).     

A stochastic theory of adiabatic invariance

LetI be a set of invariants for a system of differential equations with an ordero() vector field. When order perturbations of zero mean are added to the system we show that, under suitable regularity and ergodicity conditions,I becomes an adiabatic invariant with maximal variations of order one on time scales of order 1/². In the stochastically perturbed case,I behaves asymptotically (for small ) like a diffusion process on 1/² time scales. The results also apply to an interesting class of deterministic perturbations. This study extends the results of Khas'minskii on stochastically averaged systems, as well as some of the deterministic methods of averaging, to such invariants.Supported by NSF grant DMR-8704348     

Brownian motion in magnetic field: A model for a semi-quantum stochastic process

The Brownian motion of an ensemble of charged particles in a quantizing magnetic field is considered in a simple quasi-classical model. The model describes random jumps between the Landau levels—corresponding to the quantized motion perpendicular to the magnetic field, and markovian fluctuations in momentum space—corresponding to the classical motion parallel to the field. A random walk dynamics is adopted for the former while an Ornstein-Uhlenbeck process is considered for the latter. The resulting Fokker-Planck equations constitute a semi-quantum stochastic process. The equations are solved and several physical aspects of the underlying system are discussed.     

Entanglement of a degenerate system in adiabatic process

Entanglement evolution of a degenerate system is studied. It is found that an entangled state becomes unentangled under an adiabatic variation of the system, and vice versa. The amount of entanglement varies with different adiabatic paths. It is concluded that the entanglement of complete set of eigenstates of a system at thermal equilibrium is independent on perturbations, but the entanglement of partial eigenstates may change with adiabatic perturbations. This can help us to control entanglement properties and is very useful in the quantum computations.     

Neuronal growth: a bistable stochastic process

The fundamentally stochastic nature of neuronal growth has hardly been addressed in neuroscience. We report on the stochastic fluctuations of a neuronal growth cone's leading edge movement, the basic step in neuronal growth. Describing the edge movement as a stochastic bistable process leads to an isotropic noise parameter that is successfully used to test the model. An analysis of growth cone motility confirms the model, and predicts that linear changes of the bistable potential, as known from stochastic filtering, result in directed growth cone translocation.     

Relations between different representations of stochastic adiabatic solutions

We present ESR-data for Gd³⁺-impurities in the heavy fermion compound CeAl₃ under hydrostatic pressure up to 17kbar. At ambient pressure andT<4K the Gd-resonance shows a strongly enhanced thermal broadening with respect to isostructural LaAl₃. With increasing pressure this slope decreases from at least 138 Oe/K (p=0) to 18 Oe/K (p=17kbar) which is almost the Korringa slope of LaAl₃. These results support the idea of an increasing Ce-spin fluctuation rate as a function of pressure.     

A model for a stochastic multiphoton absorption process of an anharmonic oscillator

A model of multiphoton absorption of i.r. laser radiation by an anharmonic, molecular, vibrational mode is discussed. The multiphoton absorption is described as a stochastic process for which the reservoir, including rotation and other vibrational modes, modulates the frequency of the active oscillator stochastically. The treatment of the nonlinear oscillator is based on quantization of a classical, driven anharmonic oscillator.     

Use and misuse of adiabatic elimination procedures for stochastic processes

The extension of the current adiabatic elimination procedures to stochastic processes in the presence of external noises is criticized. Limitations pointed out in the recent literature are traced back to preparation phase effects that are shown to be unavoidable in the absence of analytical expressions for the stationary distributions.     

Entropy production along a stochastic trajectory and an integral fluctuation theorem

For stochastic nonequilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both Delatas(tot) is shown to obey a fluctuation theorem (exp([-Deltas(tot) = 1 for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.     

Entropy majorization, thermal adiabatic theorem, and quantum phase transitions

Let a general quantum many-body system at a low temperature adiabatically cross through the vicinity of the system’s quantum critical point. We show that the system’s temperature is significantly suppressed due to both the entropy majorization theorem in quantum information science and the entropy conservation law in reversible adiabatic processes. We take the one-dimensional transverse-field Ising model and the spinless fermion system as concrete examples to show that the inverse temperature might become divergent around the systems’ critical points. Since the temperature is a measurable quantity in experiments, it can be used, via reversible adiabatic processes at low temperatures, to detect quantum phase transitions in the perspectives of quantum information science and quantum statistical mechanics.     

可逆磁化区对退磁过程的影响研究

静态磁化曲线与磁滞回线测量实验中,材料剩磁较小时(不超过2.5 mT)所需加的反向外磁场强度不易把握,很难实现完全退磁.结合原理,本文对退磁过程和可逆磁化区的关系进行了定量研究,结果表明:退磁过程所需加的最小有效反向外磁场强度与可逆磁化区到不可逆磁化区的临界外磁场强度近似相等,且该值接近材料矫顽力的0.55倍.     

Protein folding as a stochastic process

In physiological conditions globular protein molecules assume a specific native conformation uniquely determined by its amino acid sequence. Upon environmental changes the protein molecules undergo reversible unfolding (order losing) and folding (order gaining) transitions, which is similar to the first-order phase transition. Pathways of folding have been intensively studied in the hope of deciphering the code that amino acid sequences carry as to the threedimensional structure of proteins. A strongly simplifiedlattice model of proteins has been found to be a powerful theoretical tool to simulate the dynamic process of the folding and unfolding transitions. The results of the simulation indicate the existence of stochastic pathways of folding.     

The universe as a stochastic process

A stochastic radiation-filled world model is constructed in which the equation of state is perturbed with a “white noise”. The corresponding Fokker-Planck equation is solved. It turns out that the maximum probability path of the world's evolution is that of the Friedman model. However, singularities are no obstacles for the stochastic evolution, and σ² → ∞ as t → ∞.     

Entropy evolution law in a laser process

For the first time, we obtain the entropy variation law in a laser process after finding the Kraus operator of the master equation describing the laser process with the use of the entangled state representation. The behavior of entropy is determined by the competition of the gain and damping in the laser process. The evolution formula for the number of photons is also obtained.     

Modelling fluctuations of financial time series: from cascade process to stochastic volatility model

In this paper, we provide a simple, “generic” interpretation of multifractal scaling laws and multiplicative cascade process paradigms in terms of volatility correlations. We show that in this context 1/f power spectra, as recently observed in reference [23], naturally emerge. We then propose a simple solvable “stochastic volatility” model for return fluctuations. This model is able to reproduce most of recent empirical findings concerning financial time series: no correlation between price variations, long-range volatility correlations and multifractal statistics. Moreover, its extension to a multivariate context, in order to model portfolio behavior, is very natural. Comparisons to real data and other models proposed elsewhere are provided. Received 22 May 2000     

The demagnetization tensor of a cylinder

All independent components of the demagnetization tensor of a cylinder are expressed by means of tabulated functions. That allows to determine the demagnetizing field of a cylinder homogeneously magnetized along an arbitrary direction. As an example the demagnetizing field is computed over the endface of a cylinder homogeneously magnetized perpendicular to its axis of rotation.