Zwanzig Model Research Articles

Lack-of-fit reduction in non-equilibrium thermodynamics applied to the Kac–Zwanzig model

Abstract Even when microscopic particle dynamics is purely mechanistic and thus reversible, the behavior of macroscopic systems composed of those particles is irreversible. In other words, effectively irreversible behavior emerges out of purely reversible dynamics when we do not observe all degrees of freedom of the detailed dynamics. But how can we find the irreversible macroscopic evolution equations when we only know the reversible microscopic equations? Using the so-called lack-of-fit reduction, which gives the reduced evolution as a sum of Hamiltonian and gradient dynamics, we reduce the purely Hamiltonian Kac–Zwanzig model to a set of irreversible evolution equations with no fitting parameters.

Investigation of the Maier–Saupe–Zwanzig Model in the Apollonian Network

Investigation of the Maier–Saupe–Zwanzig Model in the Apollonian Network

Nontrivial Dynamic Regimes of Small (Nano-Scale) Quantum Systems

Small (but still containing many atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in-between these limits. System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system - reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).

NON-TRIVIAL DYNAMIC REGIMES OF SMALL (NANO-SCALE) QUANTUM SYSTEMS

Small (but still containing many atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins. Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior. The topic of our paper is in-between these limits. System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics. We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period. We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states. In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system - reservoir coupling matrix elements are independent of the reservoir states. We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations. The coupling results to temperature dependent broadening and decay of the echo components. Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir. We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).

Response theory and phase transitions for the thermodynamic limit of interacting identical systems.

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers–Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker–Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai–Zwanzig model and of the Bonilla–Casado–Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively.

Structure of lead silicate glasses and its correlation with photoelastic properties

Short-range structures of xPbO–(100 − x)SiO2 (x = 50, 60 and 65 mol%) glasses were studied by neutron diffraction and Reverse Monte Carlo modeling. Si–O atomic pair correlation distributions are symmetrical and show peaks in the range: 1.60–1.64 ± 0.02 A. Si4+ are tetrahedrally co-ordinated with oxygen, whereas the Pb–O and O–O atomic pair correlation distributions are broad and asymmetrical due to the existence of wide range of Pb–O and O–O distances in the glass network. The peak positions in the Pb–O atomic pair correlations shift from 2.60 ± 0.05 to 2.42 ± 0.05 A on increasing PbO concentration from 50 to 65 mol%. Pb2+ exist in PbOx (x = 3, 4, 5 and 6) structural units, and the average Pb–O coordination is constant and is in the range of 4.08 ± 0.11 to 4.14 ± 0.08. The O–Si–O bond angles distributions are broad and asymmetrical with peak values in the range of 91° to 109°, and deviate significantly from the value of 109.5° in the ideal tetrahedral structural units. The short-range structural properties of glasses i.e. the cation-oxygen coordination numbers and bond lengths were used to predict the photoelastic properties of the glasses by the Zwanziger model, and it is concluded that xPbO–(100 − x)SiO2 (x = 50, 60 and 65 mol%) glasses should exhibit the properties of zero-stress birefringence.

Photon entanglement on a chip, optical instability, and Haken–Zwanzig model

There is a growing interest in the development of chip-scale ring resonator sources for entangled photons produced in four-wave mixing processes. Therefore, theoretical accounts that describe in particular the optical instability at which entangled photons emerge and characterize the degree of entanglement by means of suitably defined entanglement functions are needed. The present study describes a quantum-mechanical model for a class of chip devices that have been developed during the past 5 years and typically involve a Kerr nonlinearity in addition to the four-wave mixing nonlinearity. It is shown that the optical instability point can be described in terms of amplitude equations of the Haken–Zwanzig type. Using positive P representation and an entanglement function recently established in the literature, it is shown that at the instability point such devices produce entangled photons with a maximal degree of entanglement. When pumping is increased the degree of entanglement decays. This holds irrespective of the Kerr nonlinearity as long as the magnitude of the Kerr nonlinearity is comparable with the susceptibility parameter characterizing the four-wave mixing effect. The main effect of the Kerr nonlinearity is to shift the instability point to higher pumping intensities.

Classical Langevin dynamics derived from quantum mechanics

The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.

Generalized Langevin Equation: An Introductory Review for Biophysicists

An introductory, pedagogical review of the generalized Langevin equation (GLE) within the classical regime is presented. It is intended to be accessible to biophysicists with an interest in molecular dynamics (MD). Section 1 presents why the equation may be of interest within biophysical modeling. A detailed elementary first principles derivation of the (multidimensional) Kac–Zwanzig model is presented. The literature is reviewed with a focus on biophysical applications and representation by Markovian stochastic differential equations. The relationship with the Mori–Zwanzig formalism is discussed. The framework of model reduction and approximation is emphasized. Some open problems are identified.

Passage through a sub-diffusing geometrical bottleneck.

The usual Kramers theory of reaction rates in condensed media predict the rate to have an inverse dependence on the viscosity of the medium, η. However, experiments on ligand binding to proteins, performed long ago, showed the rate to have η-ν dependence, with ν in the range of 0.4-0.8. Zwanzig [J. Chem. Phys. 97, 3587 (1992)] suggested a model in which the ligand has to pass through a fluctuating opening to reach the binding site. This fluctuating gate model predicted the rate to be proportional to η-1/2. More recently, experiments performed by Xie et al. [Phys. Rev. Lett. 93, 180603 (2004)] showed that the distance between two groups in a protein undergoes not normal diffusion, but subdiffusion. Hence, in this paper, we suggest and solve a generalization of the Zwanzig model, viz., passage through an opening, whose size undergoes subdiffusion. Our solution shows that the rate is proportional to η-ν with ν in the range of 0.5-1, and hence, the subdiffusion model can explain the experimental observations.

Structure of lead tellurite glasses and its relationship with stress-optic properties

Reverse Monte Carlo (RMC) simulations were applied for modelling the atomic structure of the PbO-TeO2 glasses containing 10, 15 and 20-mol% PbO. The short-range order of the lead-tellurium-oxygen network, involving pair correlation functions of PbO, TeO and OO pairs, the average coordination numbers and the three-particle bond-angle distributions were determined by the RMC analysis. The average PbO coordination is in the range of 5.66 ± 0.19 to 6.02 ± 0.30, whereas TeO coordination (NTe-O) decreases steadily from 3.55 ± 0.04 to 3.36 ± 0.03 with increase in PbO concentration from 10 to 20 mol%. Raman studies also confirmed that NTe-O decreases on increasing PbO concentration, the latter acts as a modifier and disrupts the local tellurite speciation. The structural parameters are used in the Zwanziger model of stress-optic response to predict that PbO-TeO2 glasses should exhibit negative stress birefringence, a result which is in agreement with the experimental findings reported in the literature.

Capillary nematisation of colloidal rods in confinement

ABSTRACTWe confine a colloidal liquid crystal between parallel plates separated down to several times the rod length. By connecting the system to a reservoir we effectively create a grand canonical system, in which the liquid crystal displays an isotropic phase in the reservoir, but upon strong confinement becomes nematic between the parallel plates. This capillary nematisation transition can be followed down to the single particle level by means of laser scanning confocal microscopy. We compare the experimental findings to density functional theories (DFTs), within the Zwanzig model as well as a more advanced DFT, in which the effect of rod flexibility is taken into account.

Problems with vector confinement in 4d QCD

It is shown that vector confinement does not support bound state spectrum in the 4d Dirac equation. The same property is confirmed in the heavy–light and light–light QCD systems. This situation is compared with the confinement in the 2d system, which is generated by the gluon exchange. Considering the existing theories of confinement, it is shown that both the field correlator approach and the dual superconductor model ensure the scalar confinement in contrast to the Gribov–Zwanziger model, where the confining Coulomb potential does not support bound states in the Dirac equation.

Biaxial nematic stability in the rod-plate mixture with a dopant: The restricted-orientation model on the 3rd virial level

We analyze the effect of doping a binary mixture of uniaxial rod(R)- and plate(P)-like hard parallelepipeds having equal volumes and conjugated aspect ratios γR=1/γP=5. Uniaxial rods (L) or plates (T) with greater elongations γL=1/γT=10 were used as an admixture. The aim of the work is to examine this effect as a possible tool to stabilize the biaxial (Nb) phase in the P–R mixture, where Nb is known to be unstable with respect to demixing into two uniaxial nematics. The formalism to study orientational phase transitions is the off-lattice restricted-orientation (Zwanzig) model of a multi-component mixture of rectangular parallelepipeds with the EOS on the 3rd virial level. The phase biaxiality parameters of components and the second-order transition lines are located by direct minimization of the Gibbs energy. The results obtained by the examples of ternary systems show that introducing of small amounts of L or T into the reference P–R system promotes broadening of the Nb domain and may induce its stabilization. The case of L admixtures shows that an increase of the dopant particle volume results in a trend towards enhancement of the Nb phase relative stability.

Collectivity in diffusion of colloidal particles: from effective interactions to spatially correlated noise

The collectivity in the simultaneous diffusion of many particles, i.e. the interdependence of stochastic forces affecting different particles in the same solution, is a largely overlooked phenomenon with no well-established theory. Recently, we have proposed a novel type of thermodynamically consistent Langevin dynamics driven by spatially correlated noise (SCN) that can contribute to the understanding of this problem. This model draws a link between the theory of effective interactions in binary colloidal mixtures and the properties of SCN. In the current article, we review this model from the perspective of collective diffusion and generalize it to the case of multiple (N > 2) particles. Since our theory of SCN-driven Langevin dynamics has certain issues that could not be resolved within this framework, in this article we also provide another approach to the problem of collectivity. We discuss the multi-particle Mori–Zwanzig model, which is fully microscopically consistent. Indeed, we show that this model supplies a lot of information, complementary to the SCN-based approach, e.g. it predicts the deterministic dynamics of the relative distance between the particles, it provides an approximation for non-equilibrium effective interactions and predicts the collective sub-diffusion of tracers in the group. These results provide the short-range, inertial limit of the earlier model and agree with its predictions under some general conditions. In this article we also review the origin of SCN and its consequences for a variety of physical systems, with emphasis on the colloids.

Field-dependent BRST–antiBRST transformations in Yang–Mills and Gribov–Zwanziger theories

We introduce the notion of finite BRST–antiBRST transformations, both global and field-dependent, with a doublet λa, a=1,2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in Yang–Mills theories. It turns out that the finite transformations are quadratic in their parameters. At the same time, exactly as in the case of finite field-dependent BRST transformations for the Yang–Mills vacuum functional, special field-dependent BRST–antiBRST transformations, with sa-potential parameters λa=saΛ induced by a finite even-valued functional Λ and by the anticommuting generators sa of BRST–antiBRST transformations, amount to a precise change of the gauge-fixing functional. This proves the independence of the vacuum functional under such BRST–antiBRST transformations. We present the form of transformation parameters that generates a change of the gauge in the path integral and evaluate it explicitly for connecting two arbitrary Rξ-like gauges. For arbitrary differentiable gauges, the finite field-dependent BRST–antiBRST transformations are used to generalize the Gribov horizon functional h, given in the Landau gauge, and being an additive extension of the Yang–Mills action by the Gribov horizon functional in the Gribov–Zwanziger model. This generalization is achieved in a manner consistent with the study of gauge independence. We also discuss an extension of finite BRST–antiBRST transformations to the case of general gauge theories and present an ansatz for such transformations.

Exact solution of a nonresonance Zwanzig model and “quantum chaos”

We consider the quantum dynamics of the Zwanzig model in the absence of resonance of an impurity site with a reservoir and obtain analytic expressions for the evolution of the impurity site population under different initial conditions. The agreement with numerical calculations is excellent. We analyze the possibility of the existence of quantum chaos in the model.

Communication: conical intersections between vibrationally adiabatic surfaces in methanol.

A set of seven conical intersections (CI's) in methanol between vibrationally adiabatic surfaces is reported. The intersecting surfaces represent the energies of the two asymmetric CH stretch vibrations regarded as adiabatic functions of the torsion and COH bend angles. The ab initio data are well described by an extended Zwanziger and Grant (E ⊗ e) model [J. W. Zwanziger and E. R. Grant, J. Chem. Phys. 87, 2954 (1987)] that might also be regarded as an extension of the XHL model [L.-H. Xu, J. T. Hougen, and R. M. Lees, J. Mol. Spectrosc. 293-294, 38 (2013)]. The CI's illuminate the role of geometric phase in methanol. More generally, they suggest the importance of energy transfer processes localized near the CI's.

Exact Solution to the Extended Zwanzig Model for Quasi-Sigmoidal Chemically Induced Denaturation Profiles: Specific Heat and Configurational Entropy

Temperature and chemically induced denaturation comprise two of the most characteristic mechanisms to achieve the passage from the native state N to any of the unstructured states Dj in the denatured ensemble in proteins and peptides. In this work we present a full analytical solution for the configurational partition function 𝒵qs of a homopolymer chain poly-X in the extended Zwanzig model (EZM) for a quasisigmoidal denaturation profile. This solution is built up from an EZM exact solution in the case where the fraction α of native contacts follows exact linear dependence on denaturant’s concentration ζ; thus an analytical solution for 𝒵L in the case of an exact linear denaturation profile is also provided. A recently established connection between the number ν of potential nonnative conformations per residue and temperature-independent helical propensity ω complements the model in order to identify specific proteinogenic poly-X chains, where X represents any of the twenty naturally occurring aminoacid residues. From 𝒵qs, equilibrium thermodynamic potentials like entropy 𝒮 and average internal energy 〈E〉 and thermodynamic susceptibilities like specific heat C𝓋 are calculated for poly-valine (poly-V) and poly-alanine (poly-A) chains. The influence of the rate at which native contacts denature as function of ζ on thermodynamic stability is also discussed.

Gribov horizon beyond the Landau gauge

Gribov and Zwanziger proposed a modification of Yang–Mills theory in order to cure the Gribov copy problem. We employ field-dependent BRST transformations to generalize the Gribov–Zwanziger model from the Landau gauge to general Rξ gauges. The Gribov horizon functional is presented in explicit form, in both the non-local and local variants. Finally, we show how to reach any given gauge from the Landau one.