We derive a well-behaved nonlinear extension of the non-relativistic Liouville-von Neumann dynamics driven by maximal entropy production with conservation of energy and probability. The pure state limit reduces to the usual Schroedinger evolution, while mixtures evolve towards maximum entropy equilibrium states with canonical-like probability distributions on energy eigenstates. The linear, near-equilibrium limit is found to amount to an essentially exponential relaxation to thermal equilibrium; a few elementary examples are given. In addition, the modified dynamics is invariant under the time-independent symmetry group of the hamiltonian, and also invariant under the special Galilei group provided the conservation of total momentum is accounted for as well. Similar extensions can be generated for, e.g., nonextensive systems better described by a Tsallis q-entropy.
The scale-specific information content of the infinite-range, ferromagnetic Ising model is examined by means of information-theoretic measures of high-order correlations in finite-sized systems. The order-disorder transition region can be identified through the appearance of collective order in the ferromagnetic phase. In addition, it is found that near the transition, the ferromagnetic phase is marked by characteristic information oscillations at scales comparable to the system size. The amplitude of these oscillations increases with the total number of spins, so that large-scale information measures of correlations are nonanalytic in the thermodynamic limit. In contrast, correlations at scales small relative to the system size have a monotonic behavior both above and below the transition point, and a well-defined thermodynamic limit.
The accumulation of preexisting beneficial alleles in a haploid population, under selection and infrequent recombination, and in the absence of new mutation events is studied numerically by means of detailed Monte Carlo simulations. On the one hand, we confirm our previous work, in that the accumulation rate follows modified single-site kinetics, with a timescale set by an effective selection coefficient s(eff) as shown in a previous work, and we confirm the qualitative features of the dependence of s(eff) on the population size and the recombination rate reported therein. In particular, we confirm the existence of a threshold population size below which evolution stops before the emergence of best-fit individuals. On the other hand, our simulations reveal that the population dynamics is essentially shaped by the steady accumulation of pairwise sequence correlation, causing sequence congruence in excess of what one would expect from a uniformly random distribution of alleles. By sequence congruence, we understand here the opposite of genetic distance, that is, the fraction of monomorphic sites of specified allele type in a pair of genomes (individual sequences). The effective selection coefficient changes more rapidly with the recombination rate and has a higher threshold in this parameter than found in the previous work, which neglected correlation effects. We examine this phenomenon by monitoring the time dependence of sequence correlation based on a set of sequence congruence measures and verify that it is not associated with the development of linkage disequilibrium. We also discuss applications to HIV evolution in infected individuals and potential implications for drug therapy.
The self-consistent propagation of generalized D1 [coherent-product] states and of a class of gaussian density matrix generalizations is examined, at both zero and finite-temperature, for arbitrary interactions between the localized lattice (electronic or vibronic) excitations and the phonon modes. It is shown that in all legitimate cases, the evolution of D1 states reduces to the disentangled evolution of the component D2 states. The self-consistency conditions for the latter amount to conditions for decoherence-free propagation, which complement the D2 Davydov soliton equations in such a way as to lift the nonlinearity of the evolution for the on-site degrees of freedom. Although it cannot support Davydov solitons, the coherent-product ansatz does provide a wide class of exact density-matrix solutions for the joint evolution of the lattice and phonon bath in compatible systems. Included are solutions for initial states given as a product of a [largely arbitrary] lattice state and a thermal equilibrium state of the phonons. It is also shown that external pumping can produce self-consistent Frohlich-like effects. A few sample cases of coherent, albeit not solitonic, propagation are briefly discussed.
The author calls attention to previous work with related results, which has escaped scrutiny before the publication of the article "Nonlinear quantum evolution with maximal entropy production", Phys.Rev. A63, 022105 (2001).In a recently published paper [1], I propose a nonlinear extension of the nonrelativistic Liouville-von Neumann equation, which incorporates both the unitary propagation of pure quantum states and the second principle of thermodynamics. Shortly after this work was approved for publication, it was brought to my attention [2] that a closely related undertaking on the subject was reported about two decades ago by G.P. , and later also by Korsch et al. [10,11]. The present Addendum is intended to acknowledge and correct this oversight, and to provide a few brief observations on the relation between these works and the results presented in [1].In ref.[3], Beretta et al. proposed that the dynamical principle of quantum theory be replaced by a postulated nonlinear equation of motion which is, algebraically, a generalization of Eq.(24) in [1] (see also Eq.(91)). General properties of this equation are also presented in an axiomatic framework, particularly with regard to the nature of nondissipative and equilibrium states (see also [4]). Unfortunately, the applicability of the theory was hindered by a number of mathematical difficulties, e.g., by lack of a definitive proof of the positivity of the evolution. A well-behaved example was given later for the two-level system in interaction with an external field [5]. Such problems notwithstanding, it was argued [6,7] that the proposed nonlinear evolution drives the density matrix along a direction of steepest entropy ascent under given constraints, and ref.[7] provides a notable theorem on exact, generalized Onsager reciprocity, not restricted to the near-equilibrium regime. The stability of thermodynamic equilibrium states has also been discussed [8], with reference to (but not limited to the context of) the postulated nonlinear dynamics. More recently, Korsch et al. studied a family of closely related dissipative dynamics [9], and also derived explicit solutions for a driven harmonic oscillator by Lie-algebra techniques [10].Beretta's confidence in the physicality of his construction seems to find vindication after all. In ref.[1] the theory is formulated in terms of state operators γ associated to the density matrix ρ [ρ = γγ + ] and the equation of motion is derived from a variational principle which observes the principles of quantum mechanics and the fundamental laws of thermodynamics. The existence and uniqueness of the solutions for ρ follow from the equation of motion for γ, and the positivity of the evolution is guaranteed by construction. The reader can also find alternative proofs for the fundamental properties of the equation of motion, a derivation of a near-equilibrium limit, exact to first-order deviations of γ from the equilibrium state, as well as a proof of the equivalence between symmetry covariance, conservation laws and associated com...
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