235,409 research outputs found
Selected Topics in Classical Integrability
Basic notions regarding classical integrable systems are reviewed. An
algebraic description of the classical integrable models together with the zero
curvature condition description is presented. The classical r-matrix approach
for discrete and continuum classical integrable models is introduced. Using
this framework the associated classical integrals of motion and the
corresponding Lax pair are extracted based on algebraic considerations. Our
attention is restricted to classical discrete and continuum integrable systems
with periodic boundary conditions. Typical examples of discrete (Toda chain,
discrete NLS model) and continuum integrable models (NLS, sine-Gordon models
and affine Toda field theories) are also discussed.Comment: 40 pages, Latex. A few typos correcte
Time reversal of a discrete system coupled to a continuum based on non-Hermitian flip
Time reversal in quantum or classical systems described by an Hermitian
Hamiltonian is a physically allowed process, which requires in principle
inverting the sign of the Hamiltonian. Here we consider the problem of time
reversal of a subsystem of discrete states coupled to an external environment
characterized by a continuum of states, into which they generally decay. It is
shown that, by flipping the discrete-continuum coupling from an Hermitian to a
non-Hermitian interaction, thus resulting in a non unitary dynamics, time
reversal of the subsystem of discrete states can be achieved, while the
continuum of states is not reversed. Exact time reversal requires frequency
degeneracy of the discrete states, or large frequency mismatch among the
discrete states as compared to the strength of indirect coupling mediated by
the continuum. Interestingly, periodic and frequent switch of the
discrete-continuum coupling results in a frozen dynamics of the subsystem of
discrete states.Comment: 9 pages, 4 figure
Hard rod gas with long-range interactions: Exact predictions for hydrodynamic properties of continuum systems from discrete models
One-dimensional hard rod gases are explicitly constructed as the limits of
discrete systems: exclusion processes involving particles of arbitrary length.
Those continuum many-body systems in general do not exhibit the same
hydrodynamic properties as the underlying discrete models. Considering as
examples a hard rod gas with additional long-range interaction and the
generalized asymmetric exclusion process for extended particles (-ASEP),
it is shown how a correspondence between continuous and discrete systems must
be established instead. This opens up a new possibility to exactly predict the
hydrodynamic behaviour of this continuum system under Eulerian scaling by
solving its discrete counterpart with analytical or numerical tools. As an
illustration, simulations of the totally asymmetric exclusion process
(-TASEP) are compared to analytical solutions of the model and applied to
the corresponding hard rod gas. The case of short-range interaction is treated
separately.Comment: 19 pages, 8 figure
The nonlinear heat equation on W-random graphs
For systems of coupled differential equations on a sequence of W-random
graphs, we derive the continuum limit in the form of an evolution integral
equation. We prove that solutions of the initial value problems (IVPs) for the
discrete model converge to the solution of the IVP for its continuum limit.
These results combined with the analysis of nonlocally coupled deterministic
networks in [9] justify the continuum (thermodynamic) limit for a large class
of coupled dynamical systems on convergent families of graphs
Modelling the effect of gap junctions on tissue-level cardiac electrophysiology
When modelling tissue-level cardiac electrophysiology, continuum
approximations to the discrete cell-level equations are used to maintain
computational tractability. One of the most commonly used models is represented
by the bidomain equations, the derivation of which relies on a homogenisation
technique to construct a suitable approximation to the discrete model. This
derivation does not explicitly account for the presence of gap junctions
connecting one cell to another. It has been seen experimentally [Rohr,
Cardiovasc. Res. 2004] that these gap junctions have a marked effect on the
propagation of the action potential, specifically as the upstroke of the wave
passes through the gap junction.
In this paper we explicitly include gap junctions in a both a 2D discrete
model of cardiac electrophysiology, and the corresponding continuum model, on a
simplified cell geometry. Using these models we compare the results of
simulations using both continuum and discrete systems. We see that the form of
the action potential as it passes through gap junctions cannot be replicated
using a continuum model, and that the underlying propagation speed of the
action potential ceases to match up between models when gap junctions are
introduced. In addition, the results of the discrete simulations match the
characteristics of those shown in Rohr 2004. From this, we suggest that a
hybrid model -- a discrete system following the upstroke of the action
potential, and a continuum system elsewhere -- may give a more accurate
description of cardiac electrophysiology.Comment: In Proceedings HSB 2012, arXiv:1208.315
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