This is the fourth major release of the xSPDE toolbox, which solves stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and $χ^{2}$ statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. The equations can have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector component, and at either end of any interval. xSPDE has functions that can numerically solve both ordinary and partial differential stochastic equations of any type, obtaining correlations, probabilities and averages. The toolbox has a core treating stochastic differential equations, with averages, probability distributions and full error estimates. There are stochastic extensions treating applications to partial differential equations, projected equations, quantum stochastic equations, master equations and quantum phase-space simulations including Gaussian boson sampling experiments.This is the fourth major release of the xSPDE toolbox, which solves stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and $χ^{2}$ statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. The equations can have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector component, and at either end of any interval. xSPDE has functions that can numerically solve both ordinary and partial differential stochastic equations of any type, obtaining correlations, probabilities and averages. The toolbox has a core treating stochastic differential equations, with averages, probability distributions and full error estimates. There are stochastic extensions treating applications to partial differential equations, projected equations, quantum stochastic equations, master equations and quantum phase-space simulations including Gaussian boson sampling experiments.

arXiv:2303.04448Mar 8, 20230 citations

Stochastic differential equations in quantum statistical mechanics. Observables and multiple wiener integrals

Hiroshi Nakazawa

Journal of Statistical PhysicsDec 1, 19863 citations

Stochastic Processes in Classical and Quantum Physics and Engineering

Harish Parthasarathy

3.12 Group representations in relativistic image processing 3.13 Approximate solution to the wave equation in the vicinity of the event horizon of a Schwarchild black-hole 3.14 More on group theoretic image processing 3.15 A problem in quantum information theory.3.16 Hawking radiation chapter 4 Research Topics on representations with image processing applications,breathing dynamics 4.1 Remarks on induced representations and Haar integrals.4.2 On the dynamics of breathing in the presence of a mask 4.3 Plancherel formula for SL(2, R) 4.4 Some applications of the Plancherel formula for a Lie group to image processing chapter 5 Some problems in quantum information theory 5.1 Quantum information theory and quantum blackhole physics 5.2 Harmonic analysis on the Schwarz space of G = SL(2, R) 5.3 Monotonicity of quantum entropy under pinching chapter 6 6.1 Lectures on Statistical Signal Processing 6.2 Large deviation analysis of the RLS algorithm 6.3 Problem suggested by my colleague Prof. Dhananjay 6.4 Fidelity between two quantum states 6.5 Stinespring's representation of a TPCP map, ie, of a quantum operation 6.6 Approximate computation of the matrix elements of the principal and discrete series for G = SL(2, R) 6.7 Proof of the converse part of the Cq coding theorem 6.8 Proof of the direct part of the Cq coding theorem 6.9 Induced representations 6.10 Estimating non-uniform transmission line parameters and line voltage and current using the extended Kalman filter 6.11 Multipole expansion of the magnetic field produced by a time varying current distribution chapter 7 Some aspects of stochastic processes 7.1 Problem suggested by Prof.Dhananjay 7.2 Converse of the Shannon coding theorem chapter 8 Problems in electromagnetics and gravitation chapter 9 Some more aspects of quantum information theory 9.1 An expression for state detection error.9.2 Operator convex functions and operator monotone functions 9.3 A selection of matrix inequalities 9.4 Matrix Holder Inequalityand therefore (1) can be expressed asNow suppose we interpret the quantum string field X µ as being the sum of a large classical component X µ 0 (τ, σ) and a small quantum fluctuating component δX µ (τ, σ), then we can express the above exact relation as D µν (x|x ′ ) = X µ 0 (x)X ν 0 (x ′ )+ < T (δX µ (x).δX ν (x ′ )) > and further we have approximately upto first order of smallness F µ (X, X ′ , X ′′ ) = F µ (X 0 , X 0,a , X 0,b )+F µ ,1ρ (X 0 , X ′ 0 , X ′′ 0 )δX ρ +F µ ,2ρa (X, X ′ , X ′′ )δX ρ ,a +F µ ,3ρab (X, X ′ , X ′′ )δX ρ ,ab3.12 Group representations in relativistic image processing 3.13 Approximate solution to the wave equation in the vicinity of the event horizon of a Schwarchild black-hole 3.14 More on group theoretic image processing 3.15 A problem in quantum information theory.3.16 Hawking radiation chapter 4 Research Topics on representations with image processing applications,breathing dynamics 4.1 Remarks on induced representations and Haar integrals.4.2 On the dynamics of breathing in the presence of a mask 4.3 Plancherel formula for SL(2, R) 4.4 Some applications of the Plancherel formula for a Lie group to image processing chapter 5 Some problems in quantum information theory 5.1 Quantum information theory and quantum blackhole physics 5.2 Harmonic analysis on the Schwarz space of G = SL(2, R) 5.3 Monotonicity of quantum entropy under pinching chapter 6 6.1 Lectures on Statistical Signal Processing 6.2 Large deviation analysis of the RLS algorithm 6.3 Problem suggested by my colleague Prof. Dhananjay 6.4 Fidelity between two quantum states 6.5 Stinespring's representation of a TPCP map, ie, of a quantum operation 6.6 Approximate computation of the matrix elements of the principal and discrete series for G = SL(2, R) 6.7 Proof of the converse part of the Cq coding theorem 6.8 Proof of the direct part of the Cq coding theorem 6.9 Induced representations 6.10 Estimating non-uniform transmission line parameters and line voltage and current using the extended Kalman filter 6.11 Multipole expansion of the magnetic field produced by a time varying current distribution chapter 7 Some aspects of stochastic processes 7.1 Problem suggested by Prof.Dhananjay 7.2 Converse of the Shannon coding theorem chapter 8 Problems in electromagnetics and gravitation chapter 9 Some more aspects of quantum information theory 9.1 An expression for state detection error.9.2 Operator convex functions and operator monotone functions 9.3 A selection of matrix inequalities 9.4 Matrix Holder Inequalityand therefore (1) can be expressed asNow suppose we interpret the quantum string field X µ as being the sum of a large classical component X µ 0 (τ, σ) and a small quantum fluctuating component δX µ (τ, σ), then we can express the above exact relation as D µν (x|x ′ ) = X µ 0 (x)X ν 0 (x ′ )+ < T (δX µ (x).δX ν (x ′ )) > and further we have approximately upto first order of smallness F µ (X, X ′ , X ′′ ) = F µ (X 0 , X 0,a , X 0,b )+F µ ,1ρ (X 0 , X ′ 0 , X ′′ 0 )δX ρ +F µ ,2ρa (X, X ′ , X ′′ )δX ρ ,a +F µ ,3ρab (X, X ′ , X ′′ )δX ρ ,ab

CRC Press eBooksNov 1, 20220 citations

Topics in classical and quantum linear stochastic systems

Hendra I. Nurdin

Jan 1, 20078 citations

Analogue Hawking radiation, superradiance and sonic horizons in Bose-Einstein condensates

Tracy R. Slatyer

The propagation of fluctuations in the velocity potential of an inviscid, irrotational fluid is governed by a wave equation very similar to that governing the propagation of massless scalar fields in curved spacetime. This similarity allows certain fluid systems to be used as experimentally realisable models for objects that cannot be studied under controlled conditions, such as black holes and ultrarelativitsic rotating stars. The analogy applies only to the propagation of fields in a fixed background, not to the behaviour of the background itself. However, even this limited similarity has excited a great deal of interest from researchers, since it suggests that phenomena such as superradiance and Hawking radiation may have analogues in superfluids. In recent years, experimental advances in the creation and manipulation of Bose-Einstein condensates have provided researchers with very low temperature systems possessing a uniquely simple microscopic theory. Furthermore, under certain long-length-scale approximations, Bose-Einstein condensates can be described as classical irrotational inviscid fluids, and therefore can be treated as analogues of appropriate gravitational systems. We present a comprehensive and detailed critical review of the mean-field theory of Bose-Einstein condensates, the conditions under which Bose-Einstein condensates can be treated as an analogue gravity system, the construction of analogue black holes, and several derivations of Hawking radiation and analogue Hawking radiation. The interdisciplinary nature of analogue gravity means that the relevant results are widely scattered in the literature, and we consider this critical review and discussion of the relevant literature to be a central and noteworthy element of the project. We then analyse in detail a recent derivation of analogue Hawking radiation in Bose-Einstein condensates as a consequence of quantum depletion. We elucidate the methodology and the physical meaning of the results by comparison with the derivations of Hawking radiation considered previously, and discuss possible extensions and improvements. We also present an original investigation of analogue superradiance in simple vortices, with the vortex density profile chosen to mimic a realistic vortex in a Bose-Einstein condensate. We find that superradiant amplification occurs and is likely to be significant, but our long-length-scale approximation is of dubious validity near the scattering point.The propagation of fluctuations in the velocity potential of an inviscid, irrotational fluid is governed by a wave equation very similar to that governing the propagation of massless scalar fields in curved spacetime. This similarity allows certain fluid systems to be used as experimentally realisable models for objects that cannot be studied under controlled conditions, such as black holes and ultrarelativitsic rotating stars. The analogy applies only to the propagation of fields in a fixed background, not to the behaviour of the background itself. However, even this limited similarity has excited a great deal of interest from researchers, since it suggests that phenomena such as superradiance and Hawking radiation may have analogues in superfluids. In recent years, experimental advances in the creation and manipulation of Bose-Einstein condensates have provided researchers with very low temperature systems possessing a uniquely simple microscopic theory. Furthermore, under certain long-length-scale approximations, Bose-Einstein condensates can be described as classical irrotational inviscid fluids, and therefore can be treated as analogues of appropriate gravitational systems. We present a comprehensive and detailed critical review of the mean-field theory of Bose-Einstein condensates, the conditions under which Bose-Einstein condensates can be treated as an analogue gravity system, the construction of analogue black holes, and several derivations of Hawking radiation and analogue Hawking radiation. The interdisciplinary nature of analogue gravity means that the relevant results are widely scattered in the literature, and we consider this critical review and discussion of the relevant literature to be a central and noteworthy element of the project. We then analyse in detail a recent derivation of analogue Hawking radiation in Bose-Einstein condensates as a consequence of quantum depletion. We elucidate the methodology and the physical meaning of the results by comparison with the derivations of Hawking radiation considered previously, and discuss possible extensions and improvements. We also present an original investigation of analogue superradiance in simple vortices, with the vortex density profile chosen to mimic a realistic vortex in a Bose-Einstein condensate. We find that superradiant amplification occurs and is likely to be significant, but our long-length-scale approximation is of dubious validity near the scattering point.

Jan 1, 20050 citations

Input and output channels in quantum systems and quantum stochastic differential equations

Alberto Barchielli

Lecture notes in mathematicsJan 1, 198817 citations

χ^{2}

statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. The equations can have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector component, and at either end of any interval. xSPDE has functions that can numerically solve both ordinary and partial differential stochastic equations of any type, obtaining correlations, probabilities and averages. The toolbox has a core treating stochastic differential equations, with averages, probability distributions and full error estimates. There are stochastic extensions treating applications to partial differential equations, projected equations, quantum stochastic equations, master equations and quantum phase-space simulations including Gaussian boson sampling experiments.