Classical specific heat of an atomic lattice at low temperature, revisited

Perronace, Andrea; Tenenbaum, Alexander

doi:10.1103/physreve.57.100

Cited by 10 publications

(7 citation statements)

References 16 publications

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“…Experiments in computer in case of Lenard-Jones lattice 16 also confirm this result ͑simulation time up to 10 7 time steps͒; however, the temperature beyond which it occurs is almost close to the melting temperature of the lattice. On the other hand, the result of computer simulations in case of FPU model 17 shows that the specific heat always tends to its classical value at very high temperature.…”

Section: B High-temperature Specific Heatsupporting

confidence: 78%

“…This is one of the characteristic features of a nonlinear oscillation where the frequency depends on the amplitude of oscillation. The solution of the linearized version of this model and even other models 16,17 lacks this exact nonperturbative behavior of ͑0͒ as a function of A. For Im͑t͒ Ͻ KЈ / ͑0͒ , u s ͑0͒ ͑t͒ ͓=u ͑0͒ ͑t͒cos as͔ is Fourier expanded as 19 u s ͑0͒ ͑t͒ = ͚ 0 ϱ A n ͓cos͑ n t − as͒ + cos͑ n t + as͔͒, where A n = kK q n+1/2 1+q 2n+1 and n = ͑2n +1͒ ͑0͒ 2K .…”

Section: ͑20͒mentioning

confidence: 99%

“…They have used a self-consistent phonon method 14 to show that the specific heat calculated for various one-dimensional ͑1D͒ potentials agree with the classical predictions in the high-temperature limit and also predicted using the temperature-dependent phonon frequencies as measured by inelastic neutron scattering in their calculation that C p of Nb is about 2% lower than the classical value of 3Nk B and its C v decreases linearly with temperature. The computer simulations have already been carried out in Lenard-Jones lattice 16 and Fermi-Pasta-Ulam ͑FPU͒ lattice 17 to settle the issue of the behavior of specific heat at low temperature, and their measurements show that it is lower than the classical value at low enough temperature and approaches zero when the temperature goes to zero. It has also been stressed upon in Ref.…”

Section: Introductionmentioning

confidence: 99%

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Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

Mukherjee

Hossain

2008

Phys. Rev. B

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We analyze the lattice equation of motion involving terms up to third order in lattice displacement. The phenomenological arguments suggest that the force constant D 1 of the quadratic term must always be positive and the force constant B 1 of the cubic term may take either positive or negative value. The criterion for stability of the lattice provides constraint on the relative magnitudes of the three force constants. We solve the equation of motion using root mean-square spatial fluctuation approximation and obtain the seminonperturbative dispersion relation both for positive and negative B 1 . The nature of phonon density of states curves for positive B 1 show some close resemblance with the experimental observations. At very low temperature, the specific heat of this system to leading order in large positive B 1 varies as square root of temperature and it obeys Debye's T law in one dimension for small negative B 1 . At very high temperature, the specific heat may fall below or above its classical value depending on the relative magnitudes of B 1 and D 1 for B 1 Ͼ 0 and it always falls above its classical value for B 1 Ͻ 0. The lattice model with positive B 1 emerges as a good candidate for description of a monoatomic crystal.

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Section: B High-temperature Specific Heatsupporting

confidence: 78%

Section: ͑20͒mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

Mukherjee

Hossain

2008

Phys. Rev. B

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“…The next step was made in a series of papers by a group of people around Tenenbaum (see [31]). The general approach was in principle the same as that of Livi et al The relevant difference was however the choice of the subsystem, of which the energy fluctuations should be calculated.…”

mentioning

confidence: 99%

The Fermi–Pasta–Ulam problem as a challenge for the foundations of physics

Carati

Galgani

Giorgilli

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The Fermi-Pasta-Ulam (FPU) problem is discussed in connection with its physical relevance, and it is shown how apparently there exist only two possibilities: either the FPU problem is just a curiosity, or it has a fundamental role for the foundations of physics, casting a new light on the relations between classical and quantum mechanics. To this end, a short review is given of the main conceptual proposals that have been advanced. Particular emphasis is given to the perspective of a metaequilibrium scenario, which appears to be the only possible one for the FPU paradox to survive in the physically relevant case of infinitely many particles.

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“…In the FPU problem one has a classical approach, in which initial data are considered with only some of the normal modes excited (typically, the low frequency ones), and the result found is that, for low enough specific energies, up to extremely long times energy remains confined among a certain group of low frequency modes. Already in the year 1972, in a paper of Cercignani et al [17] by the title "Zero-point energy in classical non-linear mechanics" it was however proposed that even in the FPU problem one might meet with situations of the type conceived by Nernst. Numerical studies on the variant of the FPU problem with Gibbs distributed initial data came later, with results that were variously interpreted (see 18,19). Finally a case was considered, which involves a situation very similar to the one discussed here [20], where one measures the specific heat of an FPU system (with an initial Gibbs distribution), put in contact with a heat reservoir.…”

Section: Discussionmentioning

confidence: 99%

Statistical thermodynamics for metaequilibrium or metastable states

2016

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We show how statistical thermodynamics can be formulated in situations in which thermodynamics applies, while equilibrium statistical mechanics does not. A typical case is, in the words of Landau and Lifshitz, that of partial (or incomplete) equilibrium. One has a system of interest in equilibrium with the environment, and measures one of its quantities, for example its specific heat, by raising the temperature of the environment. However, within the observation time the global system settles down to a state of apparent equilibrium, so that the measured value of the specific heat is different from the equilibrium one. In such cases formulae for quantities such as the effective specific heat exist, which are provided by Fluctuation Dissipation theory. However, what is lacking is a proof that internal energy exists, i.e., that the fundamental differential form δQ−δW (difference between the heat absorbed by the system and the work performed by it) is closed. Here we show how the coefficients of the fundamental form can be expressed in such a way that the closure property of the form becomes manifest, so that the first principle is proven. We then show that the second principle too follows, and indeed as a consequence of microscopic time-reversibility. The treatment is given in a classical Hamiltonian setting. One has a global time-independent Hamiltonian system constituted by the system of interest and two auxiliary ones controlling temperature and pressure, and the occurring of a process due to a change in the thermodynamic parameters is implemented by a suitable choice of the measure for the initial data.

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Classical specific heat of an atomic lattice at low temperature, revisited

Cited by 10 publications

References 16 publications

Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

Effect of anharmonicity on the phonon density of states and specific heat of a monoatomic, one-dimensional crystal lattice

The Fermi–Pasta–Ulam problem as a challenge for the foundations of physics

Statistical thermodynamics for metaequilibrium or metastable states

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