scholarly journals Classical Casimir free energy for two Drude spheres of arbitrary radii: A plane-wave approach

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Tanja Schoger ◽  
Gert-Ludwig Ingold
Keyword(s):  
Free Energy ◽  
Plane Wave ◽  
Temperature Limit ◽  
Round Trip ◽  
High Temperature Limit ◽  
Special Cases ◽  
Exact Analytic Expression ◽  
Analytic Expression ◽  
Analytical Expressions ◽  
Capacitance Matrix

We derive an exact analytic expression for the high-temperature limit of the Casimir interaction between two Drude spheres of arbitrary radii. Specifically, we determine the Casimir free energy by using the scattering approach in the plane-wave basis. Within a round-trip expansion, we are led to consider the combinatorics of certain partitions of the round trips. The relation between the Casimir free energy and the capacitance matrix of two spheres is discussed. Previously known results for the special cases of a sphere-plane geometry as well as two spheres of equal radii are recovered. An asymptotic expansion for small distances between the two spheres is determined and analytical expressions for the coefficients are given.

scholarly journals A 4d $$ \mathcal{N} $$ = 1 Cardy Formula

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Joonho Kim ◽  
Seok Kim ◽  
Jaewon Song
Keyword(s):  
Black Holes ◽  
Free Energy ◽  
Asymptotic Behavior ◽  
High Temperature ◽  
Gauge Theory ◽  
Chemical Potential ◽  
Temperature Limit ◽  
Superconformal Index ◽  
High Temperature Limit ◽  
Cardy Formula

Abstract We study the asymptotic behavior of the (modified) superconformal index for 4d $$ \mathcal{N} $$ N = 1 gauge theory. By considering complexified chemical potential, we find that the ‘high-temperature limit’ of the index can be written in terms of the conformal anomalies 3c − 2a. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound 1/2 < a/c < 3/2 for the interacting SCFT is satisfied. We study $$ \mathcal{N} $$ N = 1 theories that are dual to AdS5 × Yp,p and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

Classical field descriptions of one-dimensional systems

1983 ◽  
Vol 61 (4) ◽  
pp. 550-563 ◽  
Author(s):  
C. Bourbonnais ◽  
L. G. Caron
Keyword(s):  
Free Energy ◽  
Temperature Limit ◽  
Energy Functional ◽  
Classical Field ◽  
Static Approximation ◽  
One Dimensional ◽  
High Temperature Limit ◽  
Amplitude Fluctuations ◽  
Field Transformation ◽  
Rigorous Treatment

In this paper we discuss the limitations of classical field treatments of one-dimensional systems in the static approximation. Two exactly solvable Hamiltonians, the ferromagnetic Ising model, and its extension to a zero-width half-filled band, are studied after their transformation to a classical field form via the Hubbard–Stratonovich identity. The more usual two-field transformation consists of using one field to describe the divergent order parameter and another to represent the nondivergent modes. The fluctuations in this latter one are usually neglected and this is shown to lead to incorrect thermodynamic behavior throughout the critical region, which is unusually large in one-dimensional systems, and even beyond to the high temperature limit. Any limited expansion of the free energy is further seen to lead to incorrect treatment of the amplitude fluctuations. A rigorous treatment of both fields is required. Alternately, a one-field transformation can assure a simpler approach although all terms in the free energy expansion must be retained. The findings are extrapolated to other known Hamiltonians: Hubbard, Peierls and spin-Peierls, and Bardeen–Cooper–Schrieffer (BCS) superconductivity. The Peierls case is examined in some detail because the usual one-field free energy functional is not obtained by a straightforward use of the Hubbard–Stratonovich transformations. As for the BCS Hamiltonian, it is seen to be in a special class because both symmetry fields are equally divergent and are automatically treated on an equal footing.

scholarly journals ENERGY EXTRACTION FROM GRAVITATIONAL COLLAPSE TO STATIC BLACK HOLES

2003 ◽  
Vol 12 (01) ◽  
pp. 121-127 ◽  
Author(s):  
REMO RUFFINI ◽  
LUCA VITAGLIANO
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Extraction Process ◽  
Maximum Efficiency ◽  
Energy Extraction ◽  
Mass Energy ◽  
Classical Level ◽  
Static Black Hole ◽  
Exact Analytic Expression ◽  
Analytic Expression

The mass-energy formula of black holes implies that up to 50% of the energy can be extracted from a static black hole. Such a result is reexamined using the recently established analytic formulas for the collapse of a shell and the expression for the irreducible mass of a static black hole. It is shown that the efficiency of energy extraction process during the formation of the black hole is linked in an essential way to the gravitational binding energy, the formation of the horizon and the reduction of the kinetic energy of implosion. Here a maximum efficiency of 50% in the extraction of the mass energy is shown to be generally attainable in the collapse of a spherically symmetric shell: surprisingly this result holds as well in the two limiting cases of the Schwarzschild and extreme Reissner–Nordström space–times. Moreover, the analytic expression recently found for the implosion of a spherical shell to an already formed black hole leads to a new exact analytic expression for the energy extraction which results in an efficiency strictly less than 100% for any physical implementable process. There appears to be no incompatibility between General Relativity and Thermodynamics at this classical level.

Maximin: a direct approach to sustainability

2006 ◽  
Vol 11 (3) ◽  
pp. 275-300 ◽  
Author(s):  
ROBERT D. CAIRNS ◽  
NGO VAN LONG
Keyword(s):  
Precautionary Principle ◽  
Direct Approach ◽  
Necessary Condition ◽  
Hyperbolic Discount ◽  
Discount Factors ◽  
Discounted Utility ◽  
Special Cases ◽  
Analytic Expression ◽  
Hotelling's Rule ◽  
Competitive Economy

We solve directly a general maximin (sustainment, intergenerational-equity) problem. Because the shadow values of a maximin problem do not correspond to the shadow values from a general discounted-utility solution, they correspond to the prices of only a very special competitive economy. Virtual discount factors for the economy arise. They do not correspond to hyperbolic discount factors. Hartwick's rule is derived and generalized naturally to take into account non-autonomous and non-deterministic features of the economy. Under uncertainty, Hartwick's rule is the analytic expression of a form of precautionary principle. Hotelling's rule is a necessary condition, but may be more complex than has been appreciated in simple models. Some interpretations of strong sustainment are special cases of weak sustainment but, paradoxically, may be more difficult to solve.

Phase-space description of a particle in a quartic double-well potential

2014 ◽  
Vol 28 (24) ◽  
pp. 1450164 ◽  
Author(s):  
Adina Ceausu-Velcescu ◽  
Paul Blaise ◽  
Yuri P. Kalmykov
Keyword(s):  
Temperature Limit ◽  
Tunneling Effect ◽  
Quantum Corrections ◽  
Perturbative Approach ◽  
High Temperature Limit ◽  
Second Moments ◽  
Perturbative Method ◽  
Classical Distribution ◽  
Double Well Potential ◽  
Space Description

The stationary Wigner functions (WFs) have been calculated for particles evolving in a quartic double-well potential V(x) = ax2/2+bx4/4(a < 0 and b > 0), at temperature T. In the high temperature limit, the results totally agree with those obtained using Wigner's perturbative method of deriving quantum corrections to the classical distribution function. Comparison with the perturbative approach allows one to establish the range of applicability of the latter. For illustration, the second moments of the position and momentum have been calculated for the double-well potential. Furthermore, the time-evolution of the WFs for a state initially located at one of the wells has been also investigated to show the tunneling effect.

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