One-dimensional classical fluid with nearest-neighbor interaction in arbitrary external field

A system of nearest-neighbor interaction on a one-dimensional lattice is investigated, which has a quasi-stationary (and position-dependent) temperature profile. The rates of heat transfer and entropy change, as well as the diffusion equation for the temperature are studied. A q-state Potts model, and its special case, a two-state Ising model, are considered as an example.

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ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493706001621 ◽

2006 ◽

Vol 06 (01) ◽

pp. 1-21 ◽

Cited By ~ 163

Author(s):

PETER W. BATES ◽

HANNELORE LISEI ◽

KENING LU

Keyword(s):

Initial Data ◽

Nearest Neighbor ◽

Invariant Set ◽

Neighbor Interaction ◽

Forward Flow ◽

Dimensional Lattice ◽

One Dimensional ◽

Nonlinear Reaction ◽

Lattice Dynamical Systems ◽

Bounded Sets

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

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Ripples in Graphene: A Variational Approach

Communications in Mathematical Physics ◽

10.1007/s00220-020-03869-z ◽

2020 ◽

Vol 379 (3) ◽

pp. 915-954

Author(s):

Manuel Friedrich ◽

Ulisse Stefanelli

Keyword(s):

Ground State ◽

Energy Minimization ◽

Variational Approach ◽

Nearest Neighbor ◽

Interaction Energies ◽

Dimensional Problem ◽

Neighbor Interaction ◽

Configurational Energy ◽

One Dimensional ◽

Math Phys

Abstract Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of Friedrich et al. (2018), rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.

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