Masuo Suzuki

Finding Exponential Product Formulas of Higher Orders

This article is based on a talk presented at a conference “Quantum Annealing and Other Optimization Methods” held at Kolkata, India on March 2–5, 2005. It will be published in the proceedings “Quantum Annealing and Other Optimization Methods” (Springer, Heidelberg) pp. 39–70. In the present article, we review the progress in the last two decades of the work on the Suzuki-Trotter decomposition, or the exponential product formula. The simplest Suzuki-Trotter decomposition, or the well-known Trotter decomposition [1–4] is given by e x(A+B) = e xA e xB + O(x 2 ), (1) where x is a parameter and A and B are arbitrary operators with some commutation relation [A, B] 6 0. Here the product of the exponential operators on the right-hand side is regarded as an approximate decomposition of the exponential operator on the left-hand side with correction terms of the second order of x. Mathematicians put Eq. (1) in the form e xA e xB = e x(A+B)+O(x 2 ) (2)

Irreversibility and entropy production in transport phenomena I

The linear response framework was established a half-century ago, but no persuasive direct derivation of entropy production has been given in this scheme. This long-term puzzle has now been solved in the present paper. The irreversible part of the entropy production in the present theory is given by (dS/dt)irr=(dU/dt)/T with the internal energy U(t) of the relevant system. Here, U(t)=〈H0〉t=TrH0ρ(t) for the Hamiltonian H0 in the absence of an external force and for the density matrix ρ(t). As is well known, we have (dS/dt)irr=0 if we use the linear-order density matrix ρlr(t)=ρ0+ρ1(t). Surprisingly, the correct entropy production is given by the second-order symmetric term ρ2(t) as (dS/dt)irr=(1/T)TrH0ρ2′(t). This is shown to agree with the ordinary expression J⋅E/T=σE2/T in the case of electric conduction for a static electric field E, using the relations TrH0ρ2′(t)=−TrH1(t)ρ1(t)=TrA⋅Eρ1(t)=J⋅E (Joule heat), which are derived from the second-order von Neumann equation iħdρ2(t)/dt=[H0,ρ2(t)]+[H1(t),ρ1(t)]. Here H1(t) denotes the partial Hamiltonian due to the external force such as H1(t)=−e∑jri⋅E≡−A⋅E in electric conduction. Thus, the linear response scheme is not closed within the first order of an external force, in order to manifest the irreversibility of transport phenomena. New schemes of steady states are also presented by introducing relaxation-type (symmetry-separated) von Neumann equations. The concept of stationary temperature Tst is introduced, which is a function of the relaxation time τr characterizing the rate of extracting heat outside from the system. The entropy production in this steady state depends on the relaxation time. A dynamical-derivative representation method to reveal the irreversibility of steady states is also proposed. The present derivation of entropy production is directly based on the first principles of using the projected density matrix ρ2(t) or more generally symmetric density matrix ρsym(t), while the previous standard argument is due to the thermodynamic energy balance. This new derivation clarifies conceptually the physics of irreversibility in transport phenomena, using the symmetry of non-equilibrium states, and this manifests the duality of current and entropy production.

Quantum annealing of the random-field Ising model by transverse ferromagnetic interactions.

We introduce transverse ferromagnetic interactions, in addition to a simple transverse field, to accelerate the convergence of quantum annealing of the random-field Ising model. The conventional approach using only the transverse-field term is known to be plagued by slow convergence when the true ground state has strong ferromagnetic characteristics for the random-field Ising model. The transverse ferromagnetic interactions are shown to improve the performance significantly in such cases. This conclusion is drawn from the analyses of the energy eigenvalues of instantaneous stationary states as well as by the very fast algorithm of Bethe-type mean-field annealing adopted to quantum systems. The present study highlights the importance of a flexible choice of the type of quantum fluctuations to achieve the best possible performance in quantum annealing. The existence of such flexibility is an outstanding advantage of quantum annealing over simulated annealing.

Irreversibility and entropy production in transport phenomena, II: Statistical–mechanical theory on steady states including thermal disturbance and energy supply

Some general aspects of nonlinear transport phenomena are discussed on the basis of two kinds of formulations obtained by extending Kubo’s perturbational scheme of the density matrix and Zubarev’s non-equilibrium statistical operator formulation. Both formulations are extended up to infinite order of an external force in compact forms and their relationship is clarified through a direct transformation. In order to make it possible to apply these formulations straightforwardly to thermal disturbance, its mechanical formulation is given (in a more convenient form than Luttinger’s formulation) by introducing the concept of a thermal field ET which corresponds to the temperature gradient and by defining its conjugate heat operator AH=∑jhjrj for a local internal energy hj of the thermal particle j. This yields a transparent derivation of the thermal conductivity κ of the Kubo form and the entropy production (dS/dt)irr=κET2/T. Mathematical aspects of the non-equilibrium density-matrix will also be discussed. In Paper I (M. Suzuki, Physica A 390 (2011)1904), the symmetry-separated von Neumann equation with relaxation terms extracting generated heat outside the system was introduced to describe the steady state of the system. In this formulation of the steady state, the internal energy 〈H0〉t is time-independent but the field energy 〈H1〉t(=−〈A〉t⋅F) decreases as time t increases. To overcome this problem, such a statistical–mechanical formulation is proposed here as includes energy supply to the system from outside by extending the symmetry-separated von Neumann equation given in Paper I. This yields a general theory based on the density-matrix formulation on a steady state with energy supply inside and heat extraction outside and consequently with both 〈H0〉t and 〈H1〉t constant. Furthermore, this steady state gives a positive entropy production.

Multivariable nonlinear analysis of foreign exchange rates

We analyze the multivariable time series of foreign exchange rates. These are price movements that have often been analyzed, and dealing time intervals and spreads between bid and ask prices. Considering dealing time intervals as event timing such as neurons’ firings, we use raster plots (RPs) and peri-stimulus time histograms (PSTHs) which are popular methods in the field of neurophysiology. Introducing special processings to obtaining RPs and PSTHs time histograms for analyzing exchange rates time series, we discover that there exists dynamical interaction among three variables. We also find that adopting multivariables leads to improvements of prediction accuracy.

First-Principle Derivation of Entropy Production in Transport Phenomena

The linear response framework was established by Kubo a half century ago, but no clear explanation of irreversibility namely entropy production has been given in this scheme. This has been now solved. The serious puzzle up to now is the following. Even using the linear response density matrix ρlr = ρ0 + ρ1(t), it has been difficult to derive the entropy production. Surprisingly, the correct entropy production is given by the second-order term ρ2(t) as . It is shown to agree with the ordinary expression JE/T = σE2/T in the case of electric conduction for a static electric field E, where σ denotes the electric conductivity expressed by the famous canonical current-current time correlation functions in equilibrium. The present article gives a review of the derivation of entropy production (M.S., Physica A 390(2011)1904-1916) based on the first-principle of using the projected density matrix ρ2(t) or more generally ρeven(t), while the previous standard argument is due to the thermodynamic energy balance. This new derivation clarifies conceptually the physics of irreversibility in transport phenomena. In general, the transport phenomena are described by the odd part ρodd(t) of the density matrix and the entropy production (namely irreversibility) is described by the even part ρeven(t). These are related to each other through the coupled equations as , where q denotes even (symmetric) or odd (antisymmetric), Pq is the projection operator to the q part of ρ(t), and 1(t) denotes the partial Hamiltonian due to the external force such as in electric conduction. The concept of a stationary temperature Tst in steady states with current (say electric current) is also proposed by using the projected and symmetry-separated von Newmann equation introduced by the present author. The entropy production of the relevant steady state depends on this stationary temperature. A mechanical formulation of thermal conduction is given by introducing a thermal field ET and its conjugate heat operator for a local internal energy hj of the thermal particle j.

Refined formulation of quantum analysis, q-derivative and exponential splitting

Quantum analysis is reformulated to clarify its essence, namely the invariance of quantum derivative for any choice of definitions of the differential df(A) satisfying the Leibniz rule. This formulation with use of the inner derivation ?A is convenient to study quantum corrections in contrast to the Feynman operator calculus. The present analysis can also be used to find a general scheme of constructing exponential product formulae of higher order. General recursive schemes are also reviewed with an emphasis to standard symmetric splitting formulae. Multiple integral representations of q-derivatives are derived using such general integral formulae of quantum derivatives as are expressed by hyperoperators. A simple explanation of the connection between quantum derivatives and q-derivative is also given.

APPLICATION OF CHAOS GAME REPRESENTATION TO NONLINEAR TIME SERIES ANALYSIS

Iterative function systems are often used for investigating fractal structures. The method is also referred as Chaos Game Representation (CGR), and is applied for representing characteristic structures of DNA sequences visually. In this paper, we proposed an original way of plotting CGR to easily confirm the property of the temporal evaluation of a time series. We also showed existence of spurious characteristic structures of time series, if we carelessly applied the CGR to real time series. We revealed that the source of spurious identification came from non-uniformity of the frequency histograms of the time series, which is often the case of analyzing real time series. We also showed how to avoid such spurious identification by applying the method of surrogate data and introducing conditional probabilities of the time series.

Topological and Ghost Interaction Methods in Equilibrium and Non-Equilibrium Systems

A topological interaction method is proposed to study boundary correlation functions and spontaneous magnetization by introducing a topological interaction J to connect spins at two opposite boundaries and by taking the limit J → 0. Conceptual relations between the present scheme and the ghost spin method are discussed. The present paper gives a general starting point for subsequent papers on the applications of the present idea. This method is suggested to be applicable to studying boundary-boundary spin correlations, boundary critical dynamics and non-equilibrium stationary flows. Nonlinear mesoscopic phenomena in infinite systems are quite interesting to study theoretically as well as experimentally. Boundary or surface phase transitions and critical phenomena are still difficult to study rigorously even in low dimensions. A topological treatment is proposed 1) to study spontaneous symmetry breaking (SSB), particularly to calculate the spontaneous magnetization mb at the boundary of the relevant system and boundary-boundary correlation functions. In

Macroscopic Order Formation,Inflation Mechanism and Entropy Change

In 1976, the author developed the scaling theory of order formation from unstable states by extending Einstein’s linear theory of Brownian motion to nonlinear unstable cases. At the first stage, an exponential growing has been shown to be dominant, namely the initial microscopic fluctuation grows rapidly upto the macroscopic order around the onset time in the scaling regime, where the nonlinearity of the relevant system plays an essential role to stabilize the system. The author found the synergetic effect (or synergism) of the initial fluctuation, random noise and nonlinearity to the formation of macroscopic order. This scaling theory of a single macrovariable or order parameter has been extended to an infinite number of order parameters. The entropy change or entropy production is also discussed from a new point of view, namely from the symmetry of the non-equilibrium density-matrix, using the von Neumann equation. The time derivative of the entropy production for general transport phenomena is also given and consequently this formula yields microscopically the principle of minimum entropy production in the linear response scheme of transport phenomena. Furthermore, we propose here a new principle of minimum dissipation for nonlinear transport phenomena such as nonlinear electric circuits whose resistances depend on currents.