On the convexity of geometric functional of level for solutions of certain elliptic partial differential equations

Laurence, Peter

doi:10.1007/bf00945002

Cited by 22 publications

(36 citation statements)

References 12 publications

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“…In order to derive the temperature in the microcanonical ensemble, according to the definition T (E) = (∂S(E)/∂E) −1 , we shall use the following generalization [18] of the Federer-Laurence derivation formula [9][10][11][12][13]. The flux Φ ξ with a non-vanishing component in the direction orthogonal to the constant energy hypersurfaces, but tangent to the level hyper-surfaces of V , can be defined by the vector ξ = n H − (n H · n V )n V , where n H = ▽H/ ▽ H and n V = ▽V / ▽ V .…”

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confidence: 99%

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

Franzosi

2011

J Stat Phys

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We consider a generic classical many particle system described by an autonomous Hamiltonian H(x 1 , . . . , x N +2 ) which, in addition, has a conserved quantity V (x 1 , . . . , x N +2 ) = v, so that the Poisson bracket {H, V } vanishes. We derive in detail the microcanonical expressions for entropy and temperature. We show that both of these quantities depend on multidimensional integrals over sub-manifolds given by the intersection of the constant energy hyper-surfaces with those defined by V (x 1 , . . . , x N +2 ) = v. We show that temperature and higher order derivatives of entropy are microcanonical observable that, under the hypothesis of ergodicity, can be calculated as time averages of suitable functions. We derive the explicit expression of the function that gives the temperature.

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confidence: 99%

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

Franzosi

2011

J Stat Phys

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“…In [7], among other results, in the case of the Laplace equation on a plane convex annulus, certain isoperimetric inequalities, which imply that I(a) is log-convex, are proved. These inequalities were successively generalized and extended in [5] to the star-shaped case, and in [3] for equation (4.1) and a general annular domain. Here, we shall generalize these results to general dimension, that is, we shall give what we think is their natural extension for equation (4.1).…”

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confidence: 99%

“…All arguments in this proof are to be intended almost everywhere with respect to ae(a 1 5 a 2 ). Set A # 1, and let us recall two formulae expressing the derivatives of £(oc) as surface integrals (see [5]):…”

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On Symmetric Invariants of Level Surfaces Near Regular Points

Magnanini

1992

Bulletin of the London Mathematical Society

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“…[36]. By using by the Federer-Laurence derivation formula [39,40,42,43], in the case of the proposed entropy we get…”

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confidence: 97%

Microcanonical entropy for classical systems

Franzosi¹

2018

Physica A: Statistical Mechanics and its Applications

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The entropy definition in the microcanonical ensemble is revisited. We propose a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we propose, are different for small system sizes. Thus, we conclude that the Boltzmann entropy provides a correct description for macroscopic systems whereas extremely small systems should be better described with the entropy that we propose here.

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On the convexity of geometric functional of level for solutions of certain elliptic partial differential equations

Cited by 22 publications

References 12 publications

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

Microcanonical Entropy and Dynamical Measure of Temperature for Systems with Two First Integrals

On Symmetric Invariants of Level Surfaces Near Regular Points

Microcanonical entropy for classical systems

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