scholarly journals Dissipation-driven integrable fermionic systems: from graded Yangians to exact nonequilibrium steady states

2017 ◽  
Vol 3 (4) ◽  
Author(s):  
Enej Ilievski
Keyword(s):  
Steady State ◽  
Lattice Models ◽  
State Density ◽  
Nonequilibrium Steady States ◽  
Matrix Product ◽  
Finite Dimensional ◽  
Master Equation Approach ◽  
Boundary Dissipation ◽  
Higher Rank ◽  
Far From Equilibrium

Using the Lindblad master equation approach, we investigate the structure of steady-state solutions of open integrable quantum lattice models, driven far from equilibrium by incoherent particle reservoirs attached at the boundaries. We identify a class of boundary dissipation processes which permits to derive exact steady-state density matrices in the form of graded matrix-product operators. All the solutions factorize in terms of vacuum analogues of Baxter’s Q-operators which are realized in terms of non-unitary representations of certain finite dimensional subalgebras of graded Yangians. We present a unifying framework which allows to solve fermionic models and naturally incorporates higher-rank symmetries. This enables to explain underlying algebraic content behind most of the previously-found solutions.

scholarly journals Modules or Mean-Fields?

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 552 ◽  
Author(s):  
Thomas Parr ◽  
Noor Sajid ◽  
Karl J. Friston
Keyword(s):  
Steady State ◽  
Message Passing ◽  
Mean Field Theory ◽  
Functional Anatomy ◽  
Mean Field ◽  
State Density ◽  
Stochastic Dynamical Systems ◽  
Conditional Independency ◽  
The Brain ◽  
Non Equilibrium

The segregation of neural processing into distinct streams has been interpreted by some as evidence in favour of a modular view of brain function. This implies a set of specialised ‘modules’, each of which performs a specific kind of computation in isolation of other brain systems, before sharing the result of this operation with other modules. In light of a modern understanding of stochastic non-equilibrium systems, like the brain, a simpler and more parsimonious explanation presents itself. Formulating the evolution of a non-equilibrium steady state system in terms of its density dynamics reveals that such systems appear on average to perform a gradient ascent on their steady state density. If this steady state implies a sufficiently sparse conditional independency structure, this endorses a mean-field dynamical formulation. This decomposes the density over all states in a system into the product of marginal probabilities for those states. This factorisation lends the system a modular appearance, in the sense that we can interpret the dynamics of each factor independently. However, the argument here is that it is factorisation, as opposed to modularisation, that gives rise to the functional anatomy of the brain or, indeed, any sentient system. In the following, we briefly overview mean-field theory and its applications to stochastic dynamical systems. We then unpack the consequences of this factorisation through simple numerical simulations and highlight the implications for neuronal message passing and the computational architecture of sentience.

scholarly journals Charge-current correlation equalities for quantum systems far from equilibrium

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Dragi Karevski ◽  
Gunter Schütz
Keyword(s):  
Linear Response ◽  
Space Dimension ◽  
Lattice Models ◽  
Quantum Systems ◽  
Translation Invariance ◽  
Response Functions ◽  
Current Correlation ◽  
Far From Equilibrium ◽  
Conserved Currents ◽  
A Current

We prove that a recently derived correlation equality between conserved charges and their associated conserved currents for quantum systems far from equilibrium [O.A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Phys. Rev. X 6, 041065 (2016)], is valid under more general conditions than assumed so far. Similar correlation identities, which in generalized Gibbs ensembles give rise to a current symmetry somewhat reminiscent of the Onsager relations, turn out to hold also in the absence of translation invariance, for lattice models, and in any space dimension, and to imply a symmetry of the non-equilibrium linear response functions.

scholarly journals A theoretical verification of the integral fluctuation theorem for accelerated colloidal systems in the long-time limit

2021 ◽  
Author(s):  
Yash Lokare
Keyword(s):  
Steady State ◽  
Time Scales ◽  
Numerical Study ◽  
Quantitative Description ◽  
Experimental Studies ◽  
Fluctuation Theorem ◽  
Nonequilibrium Steady States ◽  
Small Scale ◽  
Colloidal Systems ◽  
Long Time

A quantitative description of the violation of the second law of thermodynamics in relatively small classical systems and over short time scales comes from the fluctuation-dissipation theorem. It has been well established both theoretically and experimentally, the validity of the fluctuation theorem to small scale systems that are disturbed from their initial equilibrium states. Some experimental studies in the past have also explored the validity of the fluctuation theorem to nonequilibrium steady states at long time scales in the asymptotic limit. To this end, a theoretical and/or purely numerical model of the integral fluctuation theorem has been presented. An approximate general expression for the dissipation function has been derived for accelerated colloidal systems trapped/confined in power-law traps. Thereafter, a colloidal particle trapped in a harmonic potential (generated by an accelerating one-dimensional optical trap) and undergoing Brownian motion has been considered for the numerical study. A toy model of a quartic potential trap in addition to the harmonic trap has also been considered for the numerical study. The results presented herein show that the integral fluctuation theorem applies not only to equilibrium steady state distributions but also to nonequilibrium steady state distributions of colloidal systems in accelerated frames of reference over long time scales.

AN INTEGRABLE DECOMPOSITION OF THE SYMMETRIC MATRIX KdV EQUATION

2008 ◽  
Vol 22 (13) ◽  
pp. 1307-1315
Author(s):  
RUGUANG ZHOU ◽  
ZHENYUN QIN
Keyword(s):  
Hamiltonian Systems ◽  
Matrix Method ◽  
Kdv Equation ◽  
Symmetric Matrix ◽  
Lax Pair ◽  
Integrable Hamiltonian Systems ◽  
R Matrix ◽  
Finite Dimensional ◽  
Higher Rank ◽  
Gaudin Models

A technique for nonlinearization of the Lax pair for the scalar soliton equations in (1+1) dimensions is applied to the symmetric matrix KdV equation. As a result, a pair of finite-dimensional integrable Hamiltonian systems, which are of higher rank generalization of the classic Gaudin models, are obtained. The integrability of the systems are shown by the explicit Lax representations and r-matrix method.

On the analysis of the integral fluctuation theorem for accelerated colloidal systems in the long-time limit

2021 ◽  
Author(s):  
Yash Lokare
Keyword(s):  
Steady State ◽  
Time Scales ◽  
Numerical Study ◽  
Quantitative Description ◽  
Experimental Studies ◽  
Fluctuation Theorem ◽  
Nonequilibrium Steady States ◽  
Small Scale ◽  
Colloidal Systems ◽  
Long Time

Abstract A quantitative description of the violation of the second law of thermodynamics in relatively small classical systems and over short time scales comes from the fluctuation-dissipation theorem. It has been well established both theoretically and experimentally, the validity of the fluctuation theorem to small scale systems that are disturbed from their initial equilibrium states. Some experimental studies in the past have also explored the validity of the fluctuation theorem to nonequilibrium steady states at long time scales in the asymptotic limit. To this end, a theoretical and/or purely numerical model of the integral fluctuation theorem has been presented. An approximate general expression for the dissipation function has been derived for accelerated colloidal systems trapped/confined in power-law traps. Thereafter, a colloidal particle trapped in a harmonic potential (generated by an accelerating one-dimensional optical trap) and undergoing Brownian motion has been considered for the numerical study. A toy model of a quartic potential trap in addition to the harmonic trap has also been considered for the numerical study. The results presented herein show that the integral fluctuation theorem applies not only to equilibrium steady state distributions but also to nonequilibrium steady state distributions of colloidal systems in accelerated frames of reference over long time scales.

LATTICE ALGEBRAS AND THE HIDDEN SYMMETRY OF THE 2D ISING MODEL IN A MAGNETIC FIELD

1991 ◽  
Vol 06 (28) ◽  
pp. 5127-5153 ◽  
Author(s):  
DAN LEVY
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
External Magnetic Field ◽  
Lattice Models ◽  
Natural Generalization ◽  
Xxz Model ◽  
Affine Hecke Algebra ◽  
Finite Dimensional ◽  
Hidden Symmetry ◽  
Boundary Terms

Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.

scholarly journals Iterative solutions to the steady-state density matrix for optomechanical systems

2015 ◽  
Vol 91 (1) ◽  
Author(s):  
P. D. Nation ◽  
J. R. Johansson ◽  
M. P. Blencowe ◽  
A. J. Rimberg
Keyword(s):  
Steady State ◽  
Density Matrix ◽  
State Density ◽  
Optomechanical Systems ◽  
Iterative Solutions

scholarly journals Ribosome flow model with extended objects

2017 ◽  
Vol 14 (135) ◽  
pp. 20170128 ◽  
Author(s):  
Yoram Zarai ◽  
Michael Margaliot ◽  
Tamir Tuller
Keyword(s):  
Steady State ◽  
Production Rate ◽  
Protein Production ◽  
Flow Model ◽  
Mean Field ◽  
State Density ◽  
Transition Rates ◽  
Ribosome Density ◽  
Protein Production Rate ◽  
Unique Steady State

We study a deterministic mechanistic model for the flow of ribosomes along the mRNA molecule, called the ribosome flow model with extended objects  (RFMEO). This model encapsulates many realistic features of translation including non-homogeneous transition rates along mRNA, the fact that every ribosome covers several codons, and the fact that ribosomes cannot overtake one another. The RFMEO is a mean-field approximation of an important model from statistical mechanics called the totally asymmetric simple exclusion process with extended objects (TASEPEO). We demonstrate that the RFMEO describes biophysical aspects of translation better than previous mean-field approximations, and that its predictions correlate well with those of TASEPEO. However, unlike TASEPEO, the RFMEO is amenable to rigorous analysis using tools from systems and control theory. We show that the ribosome density profile along the mRNA in the RFMEO converges to a unique steady-state density that depends on the length of the mRNA, the transition rates along it, and the number of codons covered by every ribosome, but not on the initial density of ribosomes along the mRNA. In particular, the protein production rate also converges to a unique steady state. Furthermore, if the transition rates along the mRNA are periodic with a common period  T then the ribosome density along the mRNA and the protein production rate converge to a unique periodic pattern with period  T , that is, the model entrains to periodic excitations in the transition rates. Analysis and simulations of the RFMEO demonstrate several counterintuitive results. For example, increasing the ribosome footprint may sometimes lead to an increase in the production rate. Also, for large values of the footprint the steady-state density along the mRNA may be quite complex (e.g. with quasi-periodic patterns) even for relatively simple (and non-periodic) transition rates along the mRNA. This implies that inferring the transition rates from the ribosome density may be non-trivial. We believe that the RFMEO could be useful for modelling, understanding and re-engineering translation as well as other important biological processes.

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