Quantum computer inverting time arrow for macroscopic systems

Since Boltzmann developed the statistical theory for macroscopic thermodynamics the question has relentlessly been put forward of how time-reversibility at microscopic level is compatible with macroscopic irreversibility. Here we show that a quantum computer can efficiently simulate a macroscopic thermodynamic process with chaotic microscopic dynamics and invert the time arrow even in presence of quantum errors. In contrast, small errors in classical computer simulation of this dynamics grow exponentially with time and rapidly destroy time-reversibility. Received 31 October 2001     

Links between microscopic and macroscopic fluid mechanics

The microscopic and macroscopic versions of fluid mechanics differ qualitatively. Microscopic particles obey time-reversible ordinary differential equations. The resulting particle trajectories {q(t)} may be time-averaged or ensemble-averaged so as to generate field quantities corresponding to macroscopic variables. On the other hand, the macroscopic continuum fields described by fluid mechanics follow irreversible partial differential equations. Smooth particle methods bridge the gap separating these two views of fluids by solving the macroscopic field equations with particle dynamics that resemble molecular dynamics. Recently, nonlinear dynamics have provided some useful tools for understanding the relationship between the microscopic and macroscopic points of view. Chaos and fractals play key roles in this new understanding. Non-equilibrium phase-space averages look very different from their equilibrium counterparts. Away from equilibrium the smooth phase-space distributions are replaced by fractional-dimensional singular distributions that exhibit time irreversibility.     

Experimental studies of the transient fluctuation theorem using liquid crystals

In a thermodynamical process, the dissipation or production of entropy can only be positive or zero, according to the second law of thermodynamics. However, the laws of thermodynamics are applicable to large systems in the thermodynamic limit. Recently a fluctuation theorem, known as the transient fluctuation theorem (TFT), which generalizes the second law of thermodynamics to small systems has been proposed. This theorem has been tested in small systems such as a colloidal particle in an optical trap. We report for the first time an analogous experimental study of TFT in a spatially extended system using liquid crystals.      

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker-Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t₀ to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t₀, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker-Planck equation are also performed in a unified way in the present method.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.     

Multiscale Thermodynamics

Multiscale thermodynamics is a theory of the relations among the levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case, but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated at the microscopic, mesoscopic, and macroscopic levels. In this paper we formulate multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels.     

Irreversible thermodynamics of the steady state with uniform disequilibrium

It is demonstrated that familiar equations of contemporary irreversible thermodynamics, which are currently believed to require a foundation in statistical microscopic theory, can be derived by purely macroscopic reasoning, provided the concept of uniform disequilibrium is accepted. The consequences may be important in relieving the customary limitation of irreversible thermodynamics to near equilibrium conditions.     

Beyond the Navier–Stokes equations: Burnett hydrodynamics

This work is mainly concerned with the extension of hydrodynamics beyond the Navier–Stokes equations, a regime known as Burnett hydrodynamics. The derivation of the Burnett equations is considered from several theoretical approaches. In particular we discuss the Chapman–Enskog, Grad’s method, and Truesdell’s approach for solving the Boltzmann equation. Also, their derivation using the macroscopic approach given by extended thermodynamics is mentioned. The problems and successes of these equations are discussed and some alternatives proposed to improve them are mentioned. Comparisons of the predictions coming from the Burnett equations with experiments and/or simulations are given in order to have the necessary elements to give a critical assessment of their validity and usefulness.     

Microscopic aspects implied by the Second Law

It is conventional to try to arrive at the Boltzmann principle and the Second Law starting with the laws of dynamics at the microscopic level. In this article the opposite view is presented: Starting with the Second Law, microscopic properties are derived. A classical result of Wien is developed into a general theorem, and the possibility of deriving the Boltzmann principle as a consequence of Carnot's theorem is discussed.     

Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I

The Boltzmann equations for a formal bimolecular chemical reaction in homogeneous gas phase are transformed into an infinite system of quadratic differential equations, by expanding the distribution functions of the molecules into the Grad polynomials. The properties of these expanded Boltzmann equations reflect the macroscopic laws. In particular they enable the Onsager reciprocity relations to be derived from time-reversal invariance.     

Tanaka Theorem for Inelastic Maxwell Models

We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [29, 31]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.     

On the invariance of constitutive equations according to the kinetic theory of gases

Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.On leave from Stevens Institute of Technology, Hoboken, New Jersey 07030.     

The Coefficient of Restitution Does Not Exceed Unity

We study a classical mechanical problem in which a macroscopic ball is reflected by a non-deformable wall. The ball is modeled as a collection of classical particles bound together by an arbitrary potential, and its internal degrees of freedom are initially set to be in thermal equilibrium. The wall is represented by an arbitrary potential which is translation invariant in two directions. We then prove that the final normal momentum can exceed the initial normal momentum at most by O(■mkT), where m is the total mass of the ball, k the Boltzmann constant, and T the temperature. This implies the well-known statement in the title in the macroscopic limit where O(■mkT) is negligible. Our result may be interpreted as a rigorous demonstration of the second law of thermodynamics in a system where a macroscopic dynamics and microscopic degrees of freedom are intrinsically coupled with each other.     

Scaling theory for homogenization of the Maxwell equations

A scaling theory for homogenization of the Maxwell equations is developed upon the representation of any field as a sum of its dipole, quadrupole, and magnetic dipole moments. This representation is exact and is connected neither with multipole expansion nor with the Helmholtz theorem. A chain of hierarchical equations is derived to calculate the moments. It is shown that the resulting macroscopic fields are governed by the homogenized Maxwell equations. Generally, these fields differ from the mean values of microscopic fields.     

Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles

We introduce an elementary energy method for the Boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components. The decomposition is useful for the study of time-asymptotic stability of nonlinear waves. The wave location is determined by the macroscopic equation. The microscopic component has an equilibrating property. The coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the Chapman-Enskog expansion. Our main result is the establishment of the positivity of shock profiles for the Boltzmann equation. This is shown by the time-asymptotic approach and the maximal principle for the collision operator.The research of the first author was supported by the Institute of Mathematics, Academia Sinica, Taipei and NSC #91-2115-M-001-004. The research of the second author was supported by the SRG of City University of Hong Kong Grant #7001426.     

Numerical strategies for filtering partially observed stiff stochastic differential equations

In this paper, we present a fast numerical strategy for filtering stochastic differential equations with multiscale features. This method is designed such that it does not violate the practical linear observability condition and, more importantly, it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach. The proposed filtering algorithm comprises of a “macro-filter” that borrows ideas from the Heterogeneous Multiscale Methods and a “micro-filter” that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro-filter and macroscopic observations at asynchronous times. We will show that the proposed micro-filter is equivalent to solving an inverse problem for parameterizing differential equations. Numerically, we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions, especially when the microscopic dynamics is not mixing at all.     

Black Hole Thermodynamics and Lorentz Symmetry

Recent developments point to a breakdown in the generalized second law of thermodynamics for theories with Lorentz symmetry violation. It appears possible to construct a perpetual motion machine of the second kind in such theories, using a black hole to catalyze the conversion of heat to work. Here we describe and extend the arguments leading to that conclusion. We suggest the inference that local Lorentz symmetry may be an emergent property of the macroscopic world with origins in a microscopic second law of causal horizon thermodynamics.