On the vacuum stress induced by uniform acceleration or supporting the ether

1977 ◽  
Vol 354 (1676) ◽  
pp. 79-99 ◽  
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Scalar Fields ◽  
Black Body ◽  
Dirichlet Boundary Conditions ◽  
Massless Scalar ◽  
Perfect Conductor ◽  
Infinite Plane ◽  
Uniform Acceleration

The stresses induced in the vacuum by the uniform acceleration of an infinite plane conductor are computed for the massless scalar and electromagnetic fields. Both Dirichlet and Neumann boundary conditions are considered for the scalar field; far from the conductor it is found, independently of the boundary condition, that the vacuum stress is ‘local’ and corresponds to the absence from the vacuum of black body radiation. Approaching the conductor, the energy density in the Dirichlet case is slightly lower than the ‘local’ term, and in the Neumann case slightly higher. At very small distances it again has the same asymptotic form for both scalar fields. For the electromagnetic field the results are similar to those for the scalar field with Dirichlet boundary conditions. Far from the conductor the spectrum is again black-body, though not Planckian. In all cases the acausal nature of ‘ perfect conductor ’ boundary conditions prevents the stress tensor from being finite on the conductor.

Stationary bound states of massless scalar fields around black holes and black hole analogues

2015 ◽  
Vol 24 (09) ◽  
pp. 1542018 ◽  
Author(s):  
Carolina L. Benone ◽  
Luís C. B. Crispino ◽  
Carlos A. R. Herdeiro ◽  
Eugen Radu
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Spatial Distribution ◽  
Scalar Field ◽  
Bound States ◽  
Neumann Boundary ◽  
Mass Term ◽  
Scalar Fields ◽  
Massless Scalar ◽  
Field System

We discuss stationary bound states, a.k.a. clouds, for a massless test scalar field around Kerr black holes (BHs) and spinning acoustic BH analogues. In view of the absence of a mass term, the trapping is achieved via enclosing the BH — scalar field system in a cavity and imposing Dirichlet or Neumann boundary conditions. We discuss the variation of these bounds states with the discrete parameters that label them, as well as their spatial distribution, complementing results in our previous work [C. L. Benone, L. C. B. Crispino, C. Herdeiro and E. Radu, Phys. Rev. D91 (2015) 104038].

scholarly journals Gravitons in a Casimir box

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bronte
Keyword(s):  
Partition Function ◽  
Boundary Conditions ◽  
Chemical Potential ◽  
Finite Size ◽  
Neumann Boundary ◽  
Scalar Fields ◽  
Casimir Energy ◽  
Massless Scalar ◽  
Analytical Control ◽  
Slab Geometry

Abstract The partition function of gravitons with Casimir-type boundary conditions is worked out. The simplest box that allows one to achieve full analytical control consists of a slab geometry with two infinite parallel planes separated by a distance d. In this setting, linearized gravity, like electromagnetism, is equivalent to two free massless scalar fields, one with Dirichlet and one with Neumann boundary conditions, which in turn may be combined into a single massless scalar with periodic boundary conditions on an interval of length 2d. When turning on a chemical potential for suitably adapted spin angular momentum, the partition function is modular covariant and expressed in terms of an Eisenstein series. It coincides with that for photons. At high temperature, the result provides in closed form all sub-leading finite-size corrections to the standard (gravitational) black body result. More interesting is the low-temperature/small distance expansion where the leading contribution to the partition function is linear in inverse temperature and given in terms of the Casimir energy of the system, whereas the leading contribution to the entropy is proportional to the area and originates from gravitons propagating parallel to the plates.

scholarly journals SCALAR CASIMIR EFFECT BETWEEN TWO CONCENTRIC D-DIMENSIONAL SPHERES

2012 ◽  
Vol 27 (18) ◽  
pp. 1250094 ◽  
Author(s):  
MUSTAFA ÖZCAN
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Complex Plane ◽  
Casimir Force ◽  
Dirichlet Boundary ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Dirichlet Boundary Conditions ◽  
Massless Scalar ◽  
Massless Scalar Field

The Casimir energy for a massless scalar field between the closely spaced two concentric D-dimensional (for D>3) spheres is calculated by using the mode summation with contour integration in the complex plane of eigenfrequencies and the generalized Abel–Plana formula for evenly spaced eigenfrequency at large argument. The sign of the Casimir energy between closely spaced two concentric D-dimensional spheres for a massless scalar field satisfying the Dirichlet boundary conditions is strictly negative. The Casimir energy between (D-1)-dimensional surfaces, close to each other is regarded as interesting both by itself and as the key to describing of stability of the attractive Casimir force.

scholarly journals Vacuum Expectation Value Profiles of the Bulk Scalar Field in the Generalized Randall-Sundrum Model

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
A. Tofighi ◽  
M. Moazzen ◽  
A. Farokhtabar
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Cosmological Constant ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Brane World ◽  
Dirichlet Boundary Conditions ◽  
World Model ◽  
Expectation Value ◽  
Bulk Scalar Field

In the generalized Randall-Sundrum warped brane-world model the cosmological constant induced on the visible brane can be positive or negative. In this paper we investigate profiles of vacuum expectation value of the bulk scalar field under general Dirichlet and Neumann boundary conditions in the generalized warped brane-world model. We show that the VEV profiles generally depend on the value of the brane cosmological constant. We find that the VEV profiles of the bulk scalar field for a visible brane with negative cosmological constant and positive tension are quite distinct from those of Randall-Sundrum model. In addition we show that the VEV profiles for a visible brane with large positive cosmological constant are also different from those of the Randall-Sundrum model. We also verify that Goldberger and Wise mechanism can work under nonzero Dirichlet boundary conditions in the generalized Randall-Sundrum model.

Asymptotic behaviour of some reaction-diffusion systems modelling complex combustion on bounded domains

1993 ◽  
Vol 123 (6) ◽  
pp. 1151-1163
Author(s):  
Joel D. Avrin
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Convergence Result ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Systems ◽  
Convergence Results ◽  
Mass Fractions

SynopsisWe consider three models of multiple-step combustion processes on bounded spatial domains. Previously, steady-state convergence results have been established for these models with zero Neumann boundary conditions imposed on the temperature as well as the mass fractions. We retain here throughout the same boundary conditions on the mass fractions, but in our first set of results we establish steady-state convergence results with fixed Dirichlet boundary conditions on the temperature. Next, under certain physically reasonable assumptions, we develop, for two of the models, estimates on the decay rates of both mass fractions to zero, while for the remaining model we develop estimates on the decay rate of one concentration to zero and establish a positive lower bound on the other mass fraction. These results hold under either set of boundary conditions, but when the Dirichlet conditions are imposed on the temperature, we are able to obtain estimates on the rate of convergence of the temperature to its (generally nonconstant) steady-state. Finally, we improve the results of a previous paper by adding a temperature convergence result.

scholarly journals Parameters for the Collapse of Turbulence in the Stratified Plane Couette Flow

2018 ◽  
Vol 75 (9) ◽  
pp. 3211-3231 ◽  
Author(s):  
Ivo G. S. van Hooijdonk ◽  
Herman J. H. Clercx ◽  
Cedrick Ansorge ◽  
Arnold F. Moene ◽  
Bas J. H. van de Wiel
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Conditions ◽  
Couette Flow ◽  
Neumann Boundary ◽  
Surface Flux ◽  
Shear Capacity ◽  
Buoyancy Flux ◽  
Dirichlet Boundary Conditions ◽  
Obukhov Length

Abstract We perform direct numerical simulation of the Couette flow as a model for the stable boundary layer. The flow evolution is investigated for combinations of the (bulk) Reynolds number and the imposed surface buoyancy flux. First, we establish what the similarities and differences are between applying a fixed buoyancy difference (Dirichlet) and a fixed buoyancy flux (Neumann) as boundary conditions. Moreover, two distinct parameters were recently proposed for the turbulent-to-laminar transition: the Reynolds number based on the Obukhov length and the “shear capacity,” a velocity-scale ratio based on the buoyancy flux maximum. We study how these parameters relate to each other and to the atmospheric boundary layer. The results show that in a weakly stratified equilibrium state, the flow statistics are virtually the same between the different types of boundary conditions. However, at stronger stratification and, more generally, in nonequilibrium conditions, the flow statistics do depend on the type of boundary condition imposed. In the case of Neumann boundary conditions, a clear sensitivity to the initial stratification strength is observed because of the existence of multiple equilibriums, while for Dirichlet boundary conditions, only one statistically steady turbulent equilibrium exists for a particular set of boundary conditions. As in previous studies, we find that when the imposed surface flux is larger than the maximum buoyancy flux, no turbulent steady state occurs. Analytical investigation and simulation data indicate that this maximum buoyancy flux converges for increasing Reynolds numbers, which suggests a possible extrapolation to the atmospheric case.

On the Number of Solutions to Semilinear Boundary Value Problems

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Markus Kunze ◽  
Rafael Ortega
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Upper Bound ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

scholarly journals Synchronization for a class of generalized neural networks with interval time-varying delays and reaction-diffusion terms

2016 ◽  
Vol 21 (3) ◽  
pp. 379-399 ◽  
Author(s):  
Qintao Gan ◽  
Tielin Liu ◽  
Chang Liu ◽  
Tianshi Lv
Keyword(s):  
Neural Networks ◽  
Boundary Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Dirichlet Boundary Conditions ◽  
Linear Matrix ◽  
Time Varying ◽  
Interval Time ◽  
Generalized Neural Networks ◽  
Special Cases

In this paper, the synchronization problem for a class of generalized neural networks with interval time-varying delays and reaction-diffusion terms is investigated under Dirichlet boundary conditions and Neumann boundary conditions, respectively. Based on Lyapunov stability theory, both delay-derivative-dependent and delay-range-dependent conditions are derived in terms of linear matrix inequalities (LMIs), whose solvability heavily depends on the information of reaction-diffusion terms. The proposed generalized neural networks model includes reaction-diffusion local field neural networks and reaction-diffusion static neural networks as its special cases. The obtained synchronization results are easy to check and improve upon the existing ones. In our results, the assumptions for the differentiability and monotonicity on the activation functions are removed. It is assumed that the state delay belongs to a given interval, which means that the lower bound of delay is not restricted to be zero. Finally, the feasibility and effectiveness of the proposed methods is shown by simulation examples.

scholarly journals Non-universal Casimir Forces at Approach to Bose–Einstein Condensation of an Ideal Gas: Effect of Dirichlet Boundary Conditions

2020 ◽  
Vol 181 (3) ◽  
pp. 944-951
Author(s):  
M. Napiórkowski ◽  
J. Piasecki ◽  
J. W. Turner
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Ideal Gas ◽  
Einstein Condensate ◽  
Dirichlet Boundary Conditions ◽  
Einstein Condensation ◽  
Casimir Forces ◽  
Leading Term ◽  
Bose Einstein

Abstract We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance D. The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose–Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of thermal Casimir force between the walls is $$\sim 1/D^{2}$$ ∼ 1 / D 2 with a non-universal amplitude. The next order correction is $$\sim \ln D/D^3$$ ∼ ln D / D 3 . These observations remain in contrast with the decay law for both the periodic and Neumann boundary conditions for which the leading term is $$\sim 1/D^3$$ ∼ 1 / D 3 with a universal amplitude. We associate this discrepancy with the D-dependent positive value of the one-particle ground state energy in the case of Dirichlet boundary conditions.

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