Thermal Density Matrix Research Articles

Improved semiclassical density matrix: taming caustics.

We present a simple method to deal with caustics in the semiclassical approximation to the thermal density matrix of a particle moving on the line. For simplicity, only its diagonal elements are considered. The only ingredient we require is the knowledge of the extrema of the Euclidean action. The procedure makes use of complex trajectories, and is applied to the quartic double-well potential.

Effects of crystallinity, transcrystallinity and crystal phases of GF/PA on friction and wear mechanisms

Relationships between manufacturing processes, microstructures, mechanical and tribological properties of GF/PA6 composite were revealed in this paper by experimental investigations and theoretical analysis. The study indicated that the wear resistance was significantly influenced by the thermal history of the composite. Decreasing cooling rate during the components thermal moulding would result in enhancing approximate 28% of wear resistance and reducing almost 14% of friction in the GF/PA6 composites. A slow cooling gives the PA6 a high crystallinity and high ratio of α/γ phases which resulted in higher thermal stabilities, higher density and harder matrix in the composites than that of fast cooling. It also improves the interfacial properties and the composite hardness. Due to the thermal decomposition, abrasion and adhesion dominated the wear mechanisms, these improvements led to a higher wear resistance and higher pv service limit in the slow cooled GF/PA6 samples than that of fast cooled samples. Interfacial debonding dominated the friction mechanisms in the unlubricated pin-on-disk tests. A poor interfacial bond led to more glass fragments falling off from the GF/PA6 pins which scratched the sliding surfaces, damaged the oxidative film and resulted in severe abrasion and adhesion in the fast cooled samples, hence leading to a high friction coefficient.

Uncertainty of path integral averages at low temperature

Burghardt, Eicke, and Stolze [J. Chem. Phys. 108, 1562 (1998)] have recently presented analytical results for the coherent state path integral (CSPI) approximation to the harmonic oscillator thermal density matrix in a generalized representation. In this work, the variance of the position and momentum operators for the more common Feynman path integral approximation to the density matrix is examined and compared with the results of the generalized CSPI approximation. Both path integral approaches are found to predict minimum uncertainty states at low enough temperatures. Particular attention is given to estimates of internal energy, which can place limits upon the temperature range over which path integral approximations are valid.

Consistent Histories Approach to the Unruh Effect

Using the history projection operator (HPO) approach to consistent histories we rederive Unruh's result that an observer constantly accelerating through the Minkowski vacuum appears to be immersed in a thermal bath. We show that propositions about any symmetry of a system always form a consistent set and that the probabilities associated with such propositions are decided by their value in the initial state. We use this fact to postulate a condition on the decoherence functional in the HPO setup. Finally we show that the Unruh effect arises from the fact that the initial density matrix corresponding to the inertial vacuum can be written as a thermal density matrix in the Fock basis associated with the accelerating observer.

An initial value representation for semiclassical time-correlation functions

We derive a new initial value representation for semiclassical time-correlation functions. This derivation combines the initial value formalism developed by Miller with the stationary phase analysis of integrals over endpoint velocities developed by Xiao and Coker [J. Chem. Phys. 102, 496 (1995)] and more recently extended by Bonella, Ciccotti, and Coker [Molec. Phys. 62, 1203 (1996)]. As a result, the determination of the classical paths within the correlation function does not require “root” searches; furthermore, the thermal density matrix within this function weights the initial and not the final positions of these paths. To prevent the correlation function from being not a smooth function of time, a semiclassical phase index similar to the Maslov index is introduced. A simple numerical example is provided and possible criticisms of our approach are discussed.

The symmetrized quantum thermal flux operator

Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L2 integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.

The density matrix approach to polarized radiative transfer

The density matrix approach to polarized radiative transfer is reviewed, with particular emphasis on the physical assumptions that are at the basis of the recent developments achieved by means of this formalism. In particular it is shown that two of the basic hypotheses (the hypothesis of neglecting correlation effects between thermal velocity and density matrix, and the hypothesis of neglecting atomic polarization in the atomic ground level) are highly questionable for the description of resonance polarization -and its modifications due to the presence of a magnetic field- in spectral lines formed in the solar atmosphere.

Accurate evaluation of real and complex time propagator of a discretized space

A numerical method for the accurate evaluation of propagator in a grid space is presented. The algorithm is stable and accurate up to a microscopically significant long time ( ≈ 100 ps for proton tunneling processes). The procedure is achieved by first initializing the small time propagator to be consistent with the grid space, and then a numerical matrix multiplication scheme is adopted to yield the propagator of a future time. The method is applied to study the motion of a proton in a bistable potential. The time invariance of the quantum thermal equilibrium density matrix and the energy eigenvalues of the Hamiltonian are used to judge the goodness of the resulting propagator.

Nonequilibrium evolution of scalar fields in FRW cosmologies.

We derive the effective equations for the out of equilibrium time evolution of the order parameter and the fluctuations of a scalar field theory in spatially flat FRW cosmologies. The calculation is performed both to one loop and in a nonperturbative, self-consistent Hartree approximation. The method consists of evolving an initial functional thermal density matrix in time and is suitable for studying phase transitions out of equilibrium. The renormalization aspects are studied in detail and we find that the counterterms depend on the initial state. We investigate the high temperature expansion and show that it breaks down at long times. We also obtain the time evolution of the initial Boltzmann distribution functions, and argue that to one-loop order or in the Hartree approximation the time evolved state is a ``squeezed'' state. We illustrate the departure from thermal equilibrium by numerically studying the case of a free massive scalar field in de Sitter and radiation-dominated cosmologies. It is found that a suitably defined nonequilibrium entropy per mode increases linearly with comoving time in a de Sitter cosmology, whereas it is not a monotonically increasing function in the radiation-dominated case.

The Chandrasekhar Mass of A Gravitating Electron Crystal

AbstractThe internal structure of a white dwarf may be changed by a strong magnetic field. A local model of the electrons is constructed within a thermal density matrix formalism, essentially a Heisenberg magnetism model. This results in a matrix Fermi function which is used to construct an isothermal model of the electron crystal. The central density of the crystal is 108kg/m3 independent of the magnetic field within the plasma and therefore lower than the relativistic density, whereas this density is constant until the Fermi momentum x f = 0.3 * me * c. Chandrasekhar masses up to 1.44 * 1.4M0 are possible for polarizations of the plasma zone lower than 0.5, if the temperature is close to the Curie point, whereas the crystal itself destabilizes the white dwarf dependent on temperature.

Quantum emission from two-dimensional black holes.

We investigate Hawking radiation from two-dimensional dilatonic black holes using standard quantization techniques. In the background of a collapsing black hole solution the Bogoliubov coefficients can be exactly determined. In the regime after the black hole has settled down to an `equilibrium' state but before the backreaction becomes important these give the known result of a thermal distribution of Hawking radiation at temperature lambda/(2pi). The density matrix is computed in this regime and shown to be purely thermal. Similar techniques can be used to derive the stress tensor. The resulting expression agrees with the derivation based on the conformal anomaly and can be used to incorporate the backreaction. Corrections to the thermal density matrix are also examined, and it is argued that to leading order in perturbation theory the effect of the backreaction is to modify the Bogoliubov transformation, but not in a way that restores information lost to the black holes.

Effects of temperature on the creation of particles in a hot Friedmann universe

The creation of particles (bosons and fermions) in a hot Friedmann universe from an initial state described by a thermal density matrix is considered in the case when the effective mass of the particles depends on the temperature. It is shown that for scalar particles allowance for the temperature effects leads to a very strong suppression of the creation process. For spinor particles, allowance for the temperature effects leads to a strong suppression of the process of creation from an initial vacuum state. However if the creation of the spinor particles takes place from a state described by a thermal density matrix, then the temperature effect on the particle creation process is relatively weak.

Quantum creation of fermions in a hot universe

The creation of spinor particles in an arbitrary external field from states described by quadratic density matrices is considered. Expressions are obtained for the spectra of the created particles. It is shown that in the case of the creation of fermions (spin 1/2) in the Friedmann universe from states described by a thermal density matrix statistical effects significantly suppress the particle creation process.

Cumulant expansions for thermal density matrices and ground state logarithmic perturbation series

We express the logarithm of the thermal density matrix ρH,β (X,X′)≡〈X‖e−βH‖X′〉, where H=H0+V, as a cumulant expansion in powers of the perturbation V, by associating with {H0,‖X〉,‖X′〉} a stochastic process. If the ground state of H0 is isolated, this stochastic process is of finite memory; it then follows from the properties of cumulants that the above expansion of ln ρ is nonsecular as β→∞ (all its terms ∼β), and is thus usable down to zero temperature, unlike the direct expansion of ρ in powers of V, whose nth term ∼βn. To determine explicitly the low-temperature behavior, we apply an analysis familiar in the theory of relaxation, and obtain the form ln ρH,β(X,X′)=h(X,X′,β) +a(X)+a(X′)°−bβ, where a,b,h are cumulant expansions in powers of V; a(X), b are independent of β; and h(X,X′,β)→0 as β→∞. Comparing with ρ→〈X‖ψ0〉e−βE0〈ψ0‖X′〉 as β→∞, where ψ0 and E0 are the ground state and energy of H, we deduce E0=b, ln〈X‖ψ0〉=a(X), i.e., the Rayleigh–Schrödinger perturbation series for E0 and ln〈X‖ψ0〉 (with 〈ψ0‖ψ0〉=1) emerge as cumulant expansions. In the case of a many-body system, the properties of cumulants immediately imply linked cluster theorems for ln ρ, as well as for E0 and ln〈X‖ψ0〉.

Renormalization group studies of tunneling systems in the condensed phase

In the path integral representation of the thermal density matrix, the sum over quantum paths is equivalent to the configurational partition function sum for a system composed of long polyatomic molecules. We give an iterative (renormalization group) approach which successively reduces the number of degrees of freedom in this isomorphic classical many-body problem to a more tractable few-body problem. We give as examples a two state tunneling system coupled to an adiabatic and nonadiabatic bath and an impurity with spin coupled to conduction band electrons.

Scalar field in the early Universe: Coherent-state representation and thermal density matrix

A coherent-state representation valid even near the singularity is constructed for each mode of a quantized scalar field in a classical spatially homogeneous anisotropic background cosmology. The stress-energy tensor expectation values are computed in a coherent state and shown to be classical except for zero-point-energy terms. The self-consistent problem of a quantized scalar field in a changing background metric is discussed. The scalar field can also be described by a density matrix rather than a pure state. The density matrix is then used to determine expectation values. The density matrix need not be a thermal distribution although such a choice is reasonable in a realistic model. Temperature estimates are made using dimensional analysis.