Distribution Function of Electron Velocity Perpendicular to the Driving Force in a Uniform Nonequilibrium Steady State

A macroscopically uniform model of a two-dimensional electron system is proposed to study nonequilibrium properties of electrical conduction. By molecular dynamics simulation, the steady state distribution function $P_y$ of electron velocity in a direction perpendicular to an external driving force is calculated. An explicit form of $P_y$ is determined within the accuracy of the numerical simulation, which fits the numerical data well even in the regime where a local equilibrium description is not valid. Although the entire structure of $P_y$ is different from that of a local equilibrium distribution function, the asymptotic structure of the tails of $P_y$ in the limit of large absolute values of the velocity is identical to that of a Maxwell distribution function with a temperature which is different from that in the equilibrium state and the kinetic temperature in the steady state

An expression of excess work during transition between nonequilibrium steady states

Excess work is a non-diverging part of the work during transition between nonequilibrium steady states (NESSs). It is a central quantity in the steady state thermodynamics (SST), which is a candidate for nonequilibrium thermodynamics theory. We derive an expression of excess work during quasistatic transitions between NESSs by using the macroscopic linear response relation of NESS. This expression is a line integral of a vector potential in the space of control parameters. We show a relationship between the vector potential and the response function of NESS, and thus obtain a relationship between the SST and a macroscopic quantity. We also connect the macroscopic formulation to microscopic physics through a microscopic expression of the nonequilibrium response function, which gives a result consistent with the previous studies.Comment: 12 pages,1 figure. Title changed. To appear in J. Phys. A: Math. Theo

Superposition of Macroscopically Distinct States in Adiabatic Quantum Computation

What are the conditions for adiabatic quantum computation (AQC) to outperform classical computation? Although there exist several quantum adiabatic algorithms achieving the strong quantum speedup, the essential keys to their speedups are still unclear. Here, we investigate the connection between superpositions of macroscopically distinct states and known examples of the speedup in AQC. To formalize this notion we consider an index $p$ that quantifies a superposition of macroscopically distinct states from the asymptotic behaviors of fluctuations of additive observables. We determine this index for five examples of adiabatic algorithms exhibiting various degrees of the speedup. The results suggest that the superposition of macroscopically distinct states is an appropriate indicator of entanglement crucial to the strong quantum speedup in AQC.Comment: 15 pages, 1 figur

Equilibrium macroscopic structure revisited from spatial constraint

In classical systems, we reexamine how macroscopic structures in equilibrium state connect with spatial con- straint on the systems: e.g., volume and density as the constraint for liquids in rigid box, and crystal lattice as the constraint for crystalline solids. We reveal that in disordered states, equilibrium macroscopic structure, depend- ing on temperature and on multibody interactions in the system, is characterized by a single special microscopic structure independent of temperature and of interactions. The special microscopic structure depends only on the spatial constraint. We demonstrate the present findings providing (i) significantly efficient and systematic prediction of macroscopic structures for possible combination of constituents in multicomponent systems, and (ii) unique and accurate determination of multibody interactions in given system from measured macroscopic structure, without performing trial-and-error simulation.Comment: 5 pages. Eq. (8) is described by other variables for practical us

Microscopic Geometry Characterizes Structure/Potential-Energy Correspondence in a Thermodynamic System

Potential energy landscape (PEL) is essential to determine phase stability, reaction path, and other important physical as well as chemical properties. Whereas given PEL can reasonably determine the properties in thermodynamically equilibrium state, it is generally unclear whether a set of known property can uniquely and/or stably determines PEL, i.e., understandings of property/PEL correspondence is basically unidirectional in the current statistical mechanics. Here we make significant advance toward bidirectional bridging of this gap for classical discrete systems under many-body interactions. Our idea is to focus on characteristic microscopic geometry in configuration space for an exactly solvable system, resulting in a new, important quantity of "harmonicity in the structural degree of freedom". This quantity reasonablly characterizes which structures in equilibrium state have practically unique and stable correspondence to PEL, without requiring any thermodynamic information such as energy or temperature. The present findings will open a gate to constructing reliable PEL, where its predictive uncertainty can be a priori known. A significant role of microscopic geometry for non-interacting system should be re-emphasized in statistical mechanics.Comment: 7 pages, 4 figure

Bidirectional-Stability Breaking in Thermodynamic Average for Classical Discrete Systems

For classical systems, expectation value of macroscopic property in equilibrium state can be typically provided through thermodynamic (so-called canonical) average, where summation is taken over possible states in phase space (or in crystalline solids, it is typically approximated on cofiguration space). Although we have a number of theoretical approaches enabling to quantitatively estimate equilibrium properties by applying given potential energy surface (PES) to the thermodynamic average, it is generally unclear whether PES can be stablly, inversely determined from a given set of properties. This essentially comes from the fact that bidirectional stability characters of thermodynamic average for classical system is not sufficiently understood so far. Our recent study reveals that for classical discrete system, this property for a set of microscopic states satisfying special condition can be well-characterized by a newly-introduced concept of anharmonicity in the structural degree of freedom of D, where these states are expected to be stably inversed to underlying PES, known without any thermodynamic information. However, it is still quantitatively unclear how the bidirectional stability character is broken inside the configuration space. Here we show that the breaking in bidirectional stability for thermodynamic average is quantitatively formulated: We find that the breaking is mainly dominated by the sum of divergence and Jacobian of vector field D in configuration space, which can be fully known a priori only from geometric information of underlying lattice, without using any thermodynamic information such as energy or temperature.Comment: 4 pages,1 figur

Sum Rules and Asymptotic Behaviors for Optical Conductivity of Nonequilibrium Many-Electron Systems

For many-electron systems, we consider a nonequilibrium state (NES) that is driven by a pump field(s), which is either an optical field or a longitudinal electric field. For the differential optical conductivity describing the differential response of the NES to a probe optical field, we derive exact sum rules and asymptotic behaviors, which open wide possibilities for experiments. In deriving these results, we have also derived universal properties of general differential response functions of time-dependent NESs of general systems.Comment: 4 pages, no figures. In v2, the title has been changed to stress that the results hold irrespective of the strength of many-body interactions. More general results, Eqs.(28) and (29), have been added. In v3, a typo in the line just after Eq.(21) has been fixed, and a few equations have been displaye

Formulation of Genuine Thermodynamic Variables from Special Microscopic States

For classical discrete systems under constant composition, it has been considered that genuine thermodynamic variables such as free energy cannot be generally determined from information about a single or a few selected microscopic states. Despite this fact, we here show that Helmholtz free energy for any given composition for disordered states can be well characterized by information about a few (R+3, where R denotes number of components) specially selected microscopic states, whose structure can be known a priori without requiring any thermodynamic information. The present study is a non-trivial extension of our recently-developed theoretical approach for special microscopic states in canonical ensemble to semi-grand canonical ensemble, which additionally enables to characterize temperature dependence of other thermodynamic variables such as internal energy and entropy.Comment: 3 page

A Single Microscopic State to Characterize Ordering Tendency in Discrete Multicomponent System

Our recent study reveals that macroscopic structure in thermodynamically equilibrium state and its temperature dependence for classical discrete system can be well-characterized by a single specially-selected microscopic state (which we call projection state: PS), whose structure can be known a priori without any information about energy or temperature. Although PS can be universally constructed for any number of components R, practical application of PS to systems with R >= 3 is non-trivial compared with R = 2 (i.e., binary system). This is because (i) essentially, multicomponent system should inevitably requires linear transformation from conventional basis functions to intuitively-interpreted cluster probability basis, i.e., multiple PS energies are required to predict one chosen pair probability, leading to practically accumulating numerical errors, and (ii) additionally, explicit formula for the transformation from basis functions to pair probabilities should be required, which has been explicitly provided up to ternary (R = 3) system so far. We here derive modified formulation to directly determine probability for likeand unlike-atom pair consisting of any chosen elements by using a single PS energy, with providing explicit relationship between basis functions and pair probabilities up to quinary (R = 5) systems. We demonstrate the validity of the formulation by comparing temperature dependence of pair probabilities for multicomponent systems from thermodynamic simulations.Comment: 8 pages. Detailed derivation of the proposed formulation of pair probability is added. Application to practical ternary system is added to demonstrate the validity of the present approac

Structure of Non-Solid Matter in Equilibrium State under NVT ensemble: New Insight from Spatial Constraint

When non-solid matter (e.g., liquids or gas) is under constant volume V and density rho (e.g., in rigid box), spatial positions for their constituents are restricted by these conditions. We recently focus on the role of constraint in classical statistical thermodynamics, and find how spatial constraint connects with equilibrium properties for crystalline solids, which has not been clarified so far. The present study extend the idea to non-solid matter under NVT ensemble in classical systems. We provide explicit representation of canonical average of radial distribution function in terms of spatial constraint, which can be well characterized by a single special microscopic state on configuration space called projection state for non-solid matter. We demonstrate that the special microscopic state can be numerically constructed for a finite number of particles.Comment: 4 pages, 4 figure