Nonequilibrium Invariant Measure under Heat Flow

Delfini, Luca; Lepri, Stefano; Livi, Roberto; Politi, Antonio

doi:10.1103/physrevlett.101.120604

Cited by 21 publications

(44 citation statements)

References 15 publications

(23 reference statements)

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“…On the other hand, numerical simulations of oscillator chains, including FPU chains, give various exponents [6,10,11,12,13] for different systems, often slightly higher than 1/3. This seems consistent with early results from mode-coupling theory (MCT), which predict a heat conductivity exponent of α = 2/5 [10,11,12], although recent MCT analyses predict exponents that depend on the leading nonlinearity [13,21] and the extent of transverse motion [12]. The apparent agreement between the numerical and MCT results has led to speculation that there may be two (or more) universality classes with different exponents [12,13].…”

Section: Pacs Numberssupporting

confidence: 75%

“…The alternative, that α for the FPU-β chain will reverse its change with N and revert to ≈ 0.4, seems unlikely. Indeed, the latest MCT results [21] obtain α = 0.5 for even potentials (including the quartic model and the FPU-β model) which is quite far from earlier [13] and our numerical results. Nevertheless, it is unclear why the pure quartic system should need exceptionally large N. A final resolution of the issue requires an analytical demonstration of an error in one of the competing methods.…”

Section: Pacs Numbersmentioning

confidence: 45%

“…The apparent agreement between the numerical and MCT results has led to speculation that there may be two (or more) universality classes with different exponents [12,13]. The most recent MCT analysis [13,21] predicts that α = 1/3 is restricted to even potentials V (z), which is why we have studied the FPU-β model here.…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Conversely, slight changes can improve the convergence to α = 1/3 as seen by adding a cubic term [13,21], transverse motion [12], or collisions [11,22]. Recent simulations of nanotubes obtain α = 1/3 [25].…”

Section: Pacs Numbersmentioning

confidence: 99%

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Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

2007

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It is shown numerically that for Fermi Pasta Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N 1/3 , relevant for the much debated question of the universality of one dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained. PACS numbers:It has long been established [1] that Fermi Pasta Ulam (FPU) chains (one dimensional chains of particles with anharmonic forces between them) may not be able to achieve thermal equilibrium with seemingly reasonable initial conditions. This has interesting implications for the ergodic hypothesis, at the foundation of statistical mechanics. The results of Ref.[1] led to the discovery of solitons in continuum versions [2], and eventually an understanding of chaotic dynamics. With regard to the initial results, a vast body of work has established [3,4] that quasi-periodic solutions exist below an energy threshold, above which the system does equilibrate.Although all this work has been with closed boundary conditions, the study of FPU chains has expanded to include heat bath boundary conditions: with slightly different temperatures imposed at the two ends, the thermal conductivity can be measured and shows anomalous properties [5,6]. Since the baths at the ends are at different temperatures, the concept of equilibrium has to be extended: a local temperature (LT) that varies smoothly along the chain has to be defined. Surprisingly, this has not been fully investigated, even though the discussion of heat conductivity is in terms of Fourier's law [5] which is meaningless if the local temperature is ill-behaved. When the heat baths at the two ends have equal temperatures, one can prove analytically that the only possible steady state is the one in thermal equilibrium [7].In this paper, we demonstrate for the first time that, for FPU chains with heat bath boundary conditions, the LT behaves paradoxically on small scales. In particular, we show through numerical simulations that for FPU-β chains with alternating light and heavy masses, connected to heat baths at temperatures T L and T R at the left and right end respectively (with ∆T = T L − T R sufficiently small that the system is in the linear response regime), the kinetic temperature of the particles oscillates as one moves along the chain. The ratio of amplitude of these oscillations to ∆T /N is constant as ∆T is reduced, and increases with the chain length N : in the vicinity of the 2i'th particle, the temperature difference between heavy and light particles scales approximately as δT ∼ [∆T / √ N ]f (i/N ). Thus if one were to coarse grain over a region of the order of the mean free path, the intracell temperature uncertainty O(∆T...

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Section: Pacs Numberssupporting

confidence: 75%

Section: Pacs Numbersmentioning

confidence: 45%

Section: Pacs Numbersmentioning

confidence: 99%

Section: Pacs Numbersmentioning

confidence: 99%

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Equilibration and Universal Heat Conduction in Fermi-Pasta-Ulam Chains

2007

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“…Recently, following a principal component analysis, we have numerically found that, in the basis identified by the eigenvectors of the covariance matrix, the nonequilibrium invariant measure of this model can be effectively expressed as the product of independent distributions aligned along collective modes that are spatially localized with power-law tails [20]. Moreover, several variables, such as the amplitudes of these modes, turn out to be Gaussian distributed.…”

Section: Introductionmentioning

confidence: 99%

A stochastic model of anomalous heat transport: analytical solution of the steady state

Lepri¹,

Mejía‐Monasterio²,

Politi³

2008

J. Phys. A: Math. Theor.

127

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Abstract. We consider a one-dimensional harmonic crystal with conservative noise, in contact with two stochastic Langevin heat baths at different temperatures. The noise term consists of collisions between neighbouring oscillators that exchange their momenta, with a rate γ. The stationary equations for the covariance matrix are exactly solved in the thermodynamic limit (N → ∞). In particular, we derive an analytical expression for the temperature profile, which turns out to be independent of γ. Moreover, we obtain an exact expression for the leading term of the energy current, which scales as 1/ √ γN . Our theoretical results are finally found to be consistent with the numerical solutions of the covariance matrix for finite N .

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