Numerical Verification of the Fluctuation-Dissipation Theorem for Isolated Quantum Systems.

Fluctuation-dissipation and flow reversal theorems are used to study long-range correlation of thermal phonons in a stationary heterogeneous mechanical system comprised of arbitrary inhomogeneous fluid flow and anisotropic solid. At thermal equilibrium, with an appropriate choice of physical observables to characterize thermal fluctuations within the fluid and within the solid, the general integral expression for the two-point correlation function of the fluctuations reduces to a linear combination of deterministic Green's functions, which describe wave propagation in opposite directions between the two points. It is demonstrated that the cross-correlation of thermal noise contains as much information about the environment as can be obtained in active reciprocal transmission experiments with transceivers placed at the two points. These findings suggest a possible application of ambient noise cross-correlation to passive acoustic characterization of inhomogeneous flows in fluid-solid systems in laboratory and geophysical settings.

Finite size behavior of the asymmetric avalanche process

Brownian motion in systems of finite size

Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT. We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT-symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT-invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.

A steady state analysis of the optimization quality of a classical self-adaptive evolution strategy (ES) on a class of robust optimization problems is presented. A novel technique for calculating progress rates for nonquadratic noisy fitness landscapes is presented. This technique yields asymptotically exact results in the infinite population size limit. This technique is applied to a class of functions with noise-induced multimodality. The resulting progress rate formulas are compared with high-precision experiments. The influence of fitness resampling is considered and the steady state behavior of the ES is derived and compared with simulations. The questions whether one should sample and average fitness values and how to choose the truncation ratio are discussed giving rise to further research perspectives.

Bacteria are known to exchange genetic information by horizontal gene transfer. Since the frequency of homologous recombination depends on the similarity between the recombining segments, several studies examined whether this could lead to the emergence of subspecies. Most of them simulated fixed-size Wright-Fisher populations, in which the genetic drift should be taken into account. Here, we use nonlinear Markov processes to describe a bacterial population evolving under mutation and recombination. We consider a population structure as a probability measure on the space of genomes. This approach implies the infinite population size limit, and thus, the genetic drift is not assumed. We prove that under these conditions, the emergence of subspecies is impossible.

A simple approximate analytical expression for the direct correlation function of a hard-sphere fluid in D dimensions within the Percus-Yevick equation is proposed. The approximation exactly reproduces the well-known results for D = 1 and D = 3, while for D = 2 it compares well to the numerical results for densities not very close to the freezing density.

Integrable systems do not obey the strong eigenstate thermalization hypothesis (ETH), which has been proposed as a mechanism of thermalization in isolated quantum systems. It has been suggested that an integrable system reaches a steady state described by a generalized Gibbs ensemble (GGE) instead of thermal equilibrium. We prove that a generalized version of the strong ETH holds for noninteracting integrable systems with translation invariance. Our generalized ETH states that any pair of energy eigenstates with similar values of local conserved quantities looks similar with respect to local observables, such as local correlations. This result tells us that an integrable system relaxes to a GGE for any initial state that has subextensive fluctuations of macroscopic local conserved quantities. Contrary to the previous derivations of the GGE, it is not necessary to assume the cluster decomposition property for an initial state.

Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) ∝ 1/|J|1+α, |J| > 1, are investigated for a range of powers α. The results are compared with those for the Sherrington–Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground-state energy densities with α, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = 2/3 already well above α = 2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent ρ that describes the variations of the ground-state energy fluctuations with system size decays monotonically from its SK value for decreasing α and appears to vanish at α = 1.

We have used the ring-polymer molecular dynamics method to calculate approximate Kubo-transformed velocity autocorrelation functions and self-diffusion coefficients for low-pressure liquid para-hydrogen at temperatures of 25 and 14 K. The resulting diffusion coefficients are shown to be consistent with experimental shear viscosities and the established finite-size relation D(L) approximately = D(infinity)-2.837k(B)T6pietaL, where k(B) is the Boltzmann constant, T the absolute temperature, eta the shear viscosity, and L the length of the (cubic) simulation cell. The diffusion coefficients D(L) obtained in simulations with finite system sizes are therefore too small. However, the extrapolation to infinite system size corrects this deficiency and leads to excellent agreement with experimental results. This both demonstrates the influence of system-size effects on quantum mechanical diffusion coefficients and provides further evidence that ring-polymer molecular dynamics is an accurate as well as practical way of including quantum effects in condensed phase molecular dynamics.

We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D(alpha) on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < or = 1/2 is proportional to the Helmholtz free energy density. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.

Inspired by the models of A. Rebei and G. J. Parker and A. Rebei et. al., we study a physical model which describes the behaviour of magnetic moments in a ferromagnet. The magnetic moments are associated to 3d electrons which interact with conduction band electrons and with phonons. We study each interaction separately and then collect the results assuming that the electron-phonon interaction can be neglected. For the case of the spin-phonon interaction, we study the derivation of the equations of motion for the classical spin vector and find that the correct behaviour, as given by the Brown equation for the spin vector and the Bloch equation, using the results obtained by D. A. Garanin for the average over fluctuations of the spin vector, can be obtained in the high temperature limit. At finite temperatures we show that the Markovian approximation for the fluctuations is not correct for time scales below some thermal correlation time $\tau_{Th}$. For the case of electrons we workout a perturbative expansion of the Feynman-Vernon functional. We find the expression for the random field correlation function. The composite model (as well as the individual models) is shown to satisfy a fluctuation-dissipation theorem for all temperature regimes if the behaviour of the coupling constants of the phonon-spin interaction remains unchanged with the temperature. The equations of motion are derived.

The classical Hertz entropy is the logarithm of the volume of phase space bounded by the constant energy surface; its quantum counterpart, the quantum Hertz entropy, is $\hat S = k_B \ln \hat N$, where the quantum operator $\hat N$ specifies the number of states with energy below a given energy eigenstate. It has been recently proved that, when an isolated quantum mechanical system is driven out of equilibrium by an external driving, the change in the expectation of its quantum Hertz entropy cannot be negative, and is null for adiabatic driving. This is in full agreement with the Clausius principle. Here we test the behavior of the expectation of the quantum Hertz entropy in the case when two identical XY spin chains initially at different temperatures are quenched into a single XY chain. We observed no quantum Hertz entropy decrease. This finding further supports the statement that the quantum Hertz entropy is a proper entropy for isolated quantum systems. We further quantify how far the quenched chain is from thermal equilibrium and the temperature of the closest equilibrium.

We investigate the linear optical polarizabilities of poly(p-phenylene vinylene) oligomers using the time dependent Hartree Fock procedure. Our analysis is based on the computation of a nonlocal response function, which describes the effects of an interaction with the electromagnetic field at one carbon atom on the distribution of charges at other atoms. This provides physical insight into intramolecular charge dynamics and the anisotropic tensorial properties of the linear polarizabilities. The results are analyzed using the coupled electronic oscillator (CEO) representation, which allows the assignment of absorption peaks to distinct transitions between Hartree Fock orbitals. The size dependence of the linear absorption shows that at 10 repeat units the optical gap has almost converged to its infinite size value, and the oscillator strength of the lowest absorption line scales linearly with size. This implies that the sizes studied exceed the exciton coherence size and the resulting spectra should mimic the infinite size limit.

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.