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Lee-peng Teo - One of the best experts on this subject based on the ideXlab platform.
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Ferminoic Casimir effect between spheres
Physical Review D, 2015Co-Authors: Lee-peng TeoAbstract:We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the Order of the Casimir interaction energy is $L^{-5}$, where $L$ is the separation between the centers of the spheres. This Order is inTermediate between that of two Dirichlet spheres (of Order $L^{-3}$) and two Neumann spheres (of Order $L^{-7}$). In the small separation limit, we derive analytically the asymptotic expansion of the Casimir interaction energy up to the next-to-Leading Order Term. The Leading Term agrees with the proximity force approximation. The result for the next-to-Leading Order Term is compared to the corresponding results for scalar fields and electromagnetic fields.
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Finite temperature Casimir interaction between spheres in (D+1)-dimensional spacetime: Exact computations and asymptotic expansions
Physical Review D, 2014Co-Authors: Lee-peng TeoAbstract:We consider the finite temperature Casimir interaction between two Dirichlet spheres in $(D+1)$-dimensional Minkowski spacetime. The Casimir interaction free energy is derived from the zero temperature Casimir interaction energy via the Matsubara formalism. In the high temperature region, the Casimir interaction is dominated by the Term with zero Matsubara frequency, and it is known as the classical Term since this Term is independent of the Planck constant $\hbar$. Explicit expression of the classical Term is derived and it is computed exactly using appropriate similarity transforms of matrices. We then compute the small separation asymptotic expansion of this classical Term up to the next-to-Leading Order Term. For the remaining part of the finite temperature Casimir interaction with nonzero Matsubara frequencies, we obtain its small separation asymptotic behavior by applying certain prescriptions to the corresponding asymptotic expansion at zero temperature. This gives us a Leading Term that is shown to agree precisely with the proximity force approximation at any temperature. The next-to-Leading Order Term at any temperature is also derived and it is expressed as an infinite sum over integrals. To obtain the asymptotic expansion at the low and medium temperature regions, we apply the inverse Mellin transform techniques. In the low temperature region, we obtain results that agree with our previous work on the zero temperature Casimir interaction.
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Material dependence of Casimir interaction between a sphere and a plate: First analytic correction beyond proximity force approximation
Physical Review D, 2013Co-Authors: Lee-peng TeoAbstract:We derive analytically the asymptotic behavior of the Casimir interaction between a sphere and a plate when the distance between them, $d$, is much smaller than the radius of the sphere, $R$. The Leading Order and next-to-Leading Order Terms are derived from the exact formula for the Casimir interaction energy. They are found to depend nontrivially on the dielectric functions of the objects. As expected, the Leading Order Term coincides with that derived using the proximity force approximation. The result on the next-to-Leading Order Term complements that found by Bimonte, Emig and Kardar [Appl. Phys. Lett. \textbf{100}, 074110 (2012)] using derivative expansion. Numerical results are presented when the dielectric functions are given by the plasma model or the Drude model, with the plasma frequency (for plasma and Drude models) and relaxation frequency (for Drude model) given respectively by 9eV and 0.035eV, the conventional values used for gold metal. It is found that if plasma model is used instead of Drude model, the error in the sum of the first two Leading Terms is at most 2%, while the error in $\theta_1$, the ratio of the next-to-Leading Order Term divided by $d/R$ to the Leading Order Term, can go up to 4.5%.
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Casimir effect between two spheres at small separations
Physical Review D, 2012Co-Authors: Lee-peng TeoAbstract:We consider the Casimir interaction between two spheres at zero and finite temperature, for both scalar fields and electromagnetic fields. Of particular interest is the asymptotic expansions of the Casimir free energy when the distance between the spheres is small. The scenario where one sphere is inside the other is discussed in detail. At zero temperature, we compute analytically the Leading and the next-to-Leading Order Terms from the functional deTerminant representation of the Casimir energy. As expected, the Leading Order Term agrees with the proximity force approximation. The results for the next-to-Leading Order Terms are new. In the limit where the radius of the outer sphere goes to infinity, the results for the sphere-plane geometry are reproduced. At finite temperature, the Leading Order Term is computed and it is found to agree completely with the proximity force approximation in the medium and high temperature regions. For the scenario where two spheres are outside each other, analogous results are obtained. In the case of Dirichlet boundary conditions on both spheres, the next-to-Leading Order Term of the zero temperature Casimir energy is found to agree with that computed recently using derivative expansion.
L. P. Teo - One of the best experts on this subject based on the ideXlab platform.
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Exact classical sphere-plate Casimir interaction in (D+1)-dimensional spacetime
Physical Review D, 2014Co-Authors: L. P. TeoAbstract:We consider the high temperature limit of the Casimir interaction between a Dirichlet sphere and a Dirichlet plate due to the vacuum fluctuations of a scalar field in $(D+1)$-dimensional Minkowski spacetime. The high temperature Leading Term of the Casimir free interaction energy is known as the classical Term since it does not depend on the Planck constant $\hbar$. From the functional representation of the zero temperature Casimir interaction energy, we use Matsubara formalism to derive the classical Term. It can be expressed as a weighted sum over logarithms of deTerminants. Using similarity transforms of matrices, we re-express this classical Term as an infinite series. This series is then computed exactly using generalized Abel-Plana summation formula. From this, we deduce the short distance asymptotic expansions of the classical Casimir interaction force. As expected, the Leading Term agrees with the proximity force approximation. The next two Terms in the asymptotic expansion are also computed. It is observed that the ratio of the next-to-Leading Order Term to the Leading Order Term is proportional to the dimension of spacetime. Hence, a larger correction to the proximity force approximation is expected in spacetime with higher dimensions. This is similar to a previous result deduced for the zero temperature case.
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Casimir interaction between spheres in $\boldsymbol{(D+1)}$-dimensional Minkowski spacetime
Journal of High Energy Physics, 2014Co-Authors: L. P. TeoAbstract:We consider the Casimir interaction between two spheres in $(D+1)$-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar fields. We consider combinations of Dirichlet and Neumann boundary conditions. The TGTG formula of the Casimir interaction energy is derived. The computations of the T matrices of the two spheres are straightforward. To compute the two G matrices, known as translation matrices, which relate the hyper-spherical waves in two spherical coordinate frames differ by a translation, we generalize the operator approach employed in [IEEE Trans. Antennas Propag. \textbf{36}, 1078 (1988)]. The result is expressed in Terms of an integral over Gegenbauer polynomials. Using our expression for the Casimir interaction energy, we derive the large separation and small separation asymptotic expansions of the Casimir interaction energy. In the large separation regime, we find that the Casimir interaction energy is of Order $L^{-2D+3}$, $L^{-2D+1}$ and $L^{-2D-1}$ respectively for Dirichlet-Dirichlet, Dirichlet-Neumann and Neumann-Neumann boundary conditions, where $L$ is the center-to-center distance of the two spheres. In the small separation regime, we confirm that the Leading Term of the Casimir interaction agrees with the proximity force approximation, which is of Order $d^{-\frac{D+1}{2}}$, where $d$ is the distance between the two spheres. Another main result of this work is the analytic computations of the next-to-Leading Order Term in the small separation asymptotic expansion. This Term is computed using careful Order analysis as well as perturbation method. We find that when $D$ is large, the ratio of the next-to-Leading Order Term to the Leading Order Term is linear in $D$, indicating a larger correction at higher dimensions.
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Sphere-plate Casimir interaction in (D+1)-dimensional spacetime
Journal of Mathematical Physics, 2014Co-Authors: L. P. TeoAbstract:In this paper, we derive the formula for the Casimir interaction energy between a sphere and a plate in $(D+1)$-dimensional Minkowski spacetime. It is assumed that the scalar field satisfies the Dirichlet or Neumann boundary conditions on the sphere and the plate. As in the $D=3$ case, the formula is of TGTG type. One of our main contributions is deriving the translation matrices which express the change of bases between plane waves and spherical waves for general $D$. Using orthogonality of Gegenbauer polynomials, it turns out that the final TGTG formula for the Casimir interaction energy can be simplified to one that is similar to the $D=3$ case. To illustrate the application of the formula, both large separation and small separation asymptotic behaviors of the Casimir interaction energy are computed. The large separation Leading Term is proportional to $L^{-D+1}$ if the sphere is imposed with Dirichlet boundary condition, and to $L^{-D-1}$ if the sphere is imposed with Neumann boundary condition, where $L$ is distance from the center of the sphere to the plane. For the small separation asymptotic behavior, it is shown that the Leading Term is equal to the one obtained using proximity force approximation. The next-to-Leading Order Term is also computed using perturbation method. It is shown that when the space dimension $D$ is larger than 5, the next-to-Leading Order has sign opposite to the Leading Order Term. Moreover, the ratio of the next-to-Leading Order Term to the Leading Order Term is linear in $D$, indicating a larger correction at higher dimensions.
Hermann Schulz - One of the best experts on this subject based on the ideXlab platform.
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gluon plasma frequency the next to Leading Order Term
Nuclear Physics, 1994Co-Authors: Hermann SchulzAbstract:Abstract The longitudinal-electric oscillations of the hot gluon system are studied beyond the well-known Leading Order Term at high temperature T and small coupling g . The coefficient η in ω 2 = m 2 (1+ ηg √ N ) is calculated, where ω = ω( q = 0 ) is the long-wavelength limit of the frequency spectrum, N the number of colours and m 2 = 1 9 g 2 NT 2 . In the course of this, for the real part of the gluon self-energy, the Braatenn-Pisarski resummation programme is found to work well in all details. The coefficient η is explicitly seen to be gauge independent within the class of covariant gauges. Infrared singularities cancel as well is collinear singularities in the two-loop diagrams with both inner momenta hard. However, as it turns out, none of these two-loop contributions reaches the relative Order O( g ) under study. The minus sign in our numerical result η = −0.18 is in accord with the intuitive picture that the studied mode might soften with increasing coupling (lower temperature) until a phase transition is reached at zero-frequency. The minus sign thus exhibits the “glue” effect for the first time in a dynamical quantity of hot QCD.
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Gluon plasma frequency — the next-to-Leading Order Term
Nuclear Physics B, 1994Co-Authors: Hermann SchulzAbstract:Abstract The longitudinal-electric oscillations of the hot gluon system are studied beyond the well-known Leading Order Term at high temperature T and small coupling g . The coefficient η in ω 2 = m 2 (1+ ηg √ N ) is calculated, where ω = ω( q = 0 ) is the long-wavelength limit of the frequency spectrum, N the number of colours and m 2 = 1 9 g 2 NT 2 . In the course of this, for the real part of the gluon self-energy, the Braatenn-Pisarski resummation programme is found to work well in all details. The coefficient η is explicitly seen to be gauge independent within the class of covariant gauges. Infrared singularities cancel as well is collinear singularities in the two-loop diagrams with both inner momenta hard. However, as it turns out, none of these two-loop contributions reaches the relative Order O( g ) under study. The minus sign in our numerical result η = −0.18 is in accord with the intuitive picture that the studied mode might soften with increasing coupling (lower temperature) until a phase transition is reached at zero-frequency. The minus sign thus exhibits the “glue” effect for the first time in a dynamical quantity of hot QCD.
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gluon plasma frequency the next to Leading Order Term
arXiv: High Energy Physics - Phenomenology, 1993Co-Authors: Hermann SchulzAbstract:The longitudinal-electric oscillations of the hot gluon system are studied beyond the well known Leading Order Term at high temperature $T$ and small coupling $g$. The coefficient $\eta$ in $\omega^2 = m^2 \, (1+ \eta \, g \wu N \, )$ is calculated, where \hbox{$\omega \equiv \omega (\vc q =0)$} is the long-wavelength limit of the frequency spectrum, $N$ the number of colours and $m^2=g^2 N T^2/9$. In the course of this, for the real part of the gluon self-energy, the Braaten-Pisarski resummation programme is found to work well in all details. The coefficient $\eta$ is explicitly seen to be gauge independent within the class of covariant gauges. Infrared singularities cancel as well as collinear singularities in the two-loop diagrams with both inner momenta hard. However, as it turns out, none of these two-loop contributions reaches the relative Order $O(g)$ under study. The minus sign in our numerical result $\; \eta = -.18 \; $ is in accord with the intuitive picture that the studied mode might soften with increasing coupling (lower temperature) until a phase transition is reached at zero-frequency. The minus sign thus exhibits the 'glue' effect for the first time in a dynamical quantity of hot QCD.
Agnessa Kovaleva - One of the best experts on this subject based on the ideXlab platform.
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Risk-Sensitive Control for Nonlinear Oscillatory Systems with Small Noise
IFAC Proceedings Volumes, 2001Co-Authors: Agnessa KovalevaAbstract:Abstract The problem of controlling a near-Hamiltonian noisy system so as to prevent it from overcoming the potential barrier is considered. An exponential risk-sensitive residence time criterion is examined as a solution of a related HJB equation. The averaged HJB equation is constructed as a first Order PDE with the coefficients dependent on the noise intensity in the Leading Order Term, though this intensity tends to zero in the original system. The Leading Order nearly optimal control is constructed as a stationary feedback with parameters dependent on the noise intensity.
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Exponential escape criteria for a quasiconservative system with small noise
2001 European Control Conference (ECC), 2001Co-Authors: Agnessa KovalevaAbstract:The problem of controlling a near-Hamiltonian noisy system so as to prevent it from overcoming the potential barrier is considered. An exponential risk-sensitive residence time criterion is examined as a solution of a related HJB equation. The averaged HJB equation is constructed as a first Order PDE with the coefficients dependent on the noise intensity in the Leading Order Term, though this intensity tends to zero in the original system. The Leading Order nearly optimal control is constructed as a stationary feedback with parameters dependent on the noise intensity.
Xin Zhou - One of the best experts on this subject based on the ideXlab platform.
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On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation
Communications on Pure and Applied Mathematics, 2004Co-Authors: Alexander Tovbis, Stephanos Venakides, Xin ZhouAbstract:We calculate the Leading-Order Term of the solution of the focusing nonlinear (cubic) Schrodinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. We utilize the Riemann-Hilbert problem formulation of the inverse scattering problem to obtain the Leading-Order Term of the solution. Error estimates are provided. © 2004 Wiley Periodicals, Inc.