Christian Gruber - Academia.edu (original) (raw)
Papers by Christian Gruber
Physica A: Statistical Mechanics and its Applications, 1996
A lattice system of electrons in the presence of an external potential and under the influence of... more A lattice system of electrons in the presence of an external potential and under the influence of a magnetic field is considered. The energy of the electrons is given by the Falicov-Kimball hamiltonian, where the ions give the external potential. The problem is to find the configurations of ions and the magnetic fluxes which minimize the energy of the electrons at zero temperature. Our results, valid for large coupling constant, show that periodic configurations with periodic fluxes (possibly inhomogeneous) appear for some values of the chemical potentials.
Journal of Statistical Physics
We investigate the evolution of a system composed of N non-interacting point particles of mass m ... more We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass MO\mathcal{O} (M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by X M (t)= L[1/2–(t)] where the function is independent of M. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.
Journal of Statistical Physics, 2002
We consider the evolution of a system composed of N non-interacting point particles of mass m in ... more We consider the evolution of a system composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adiabatic wall (the adiabatic piston). We study the thermodynamic limit for the piston where the area A of the cross-section, the mass M of the piston, and the number N of particles go
Encyclopedia of Mathematical Physics, 2006
Physica A: Statistical Mechanics and its Applications, 2002
We consider the evolution of a system composed of N non-interacting point particles of mass m in ... more We consider the evolution of a system composed of N non-interacting point particles of mass m in a container divided in two regions by a movable adiabatic wall (adiabatic piston). In this talk we discuss the thermodynamic limit where the area A of the container , the number N of particles, and the mass M of the piston go to infinity keeping A M and N M fixed. We show that in this limit the motion of the piston is deterministic. Introducing simplifying assumptions we discuss the approach to equilibrium and we illustrate the results with numerical simulations. The comparison with the case of a system with finite (A, N, M) will be presented.
Communications in Mathematical Physics, 1979
We introduce the surface tension for arbitrary spin systems and study its general properties. In ... more We introduce the surface tension for arbitrary spin systems and study its general properties. In particular we show that for a large class of systems, the surface tension is zero at high temperature. We also derive a geometrical condition for the surface tension to be zero at all temperature. For discrete spin systems this condition becomes a criterion to establish the existence of a phase transition associated with surface tension. This criterion is illustrated on several examples.
Communications in Mathematical Physics, 1975
It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Ki... more It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ρ, and for the spin correlation functions σ, are essentially equivalent for all ρ, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoίf process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.
Communications in Mathematical Physics, 1977
It is shown that for any KMS-state of a classical system of noncoincident particles, the distribu... more It is shown that for any KMS-state of a classical system of noncoincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.
Journal of Statistical Physics, 2004
The piston problem is investigated in the case where the length of the cylinder is infinite (on b... more The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio m/M is a very small parameter, where m is the mass of one particle of the gaz and M is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics.
Journal of Geodesy, 2016
The computation of spherical harmonic series in very high resolution is known to be delicate in t... more The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in spectral domain then regular grid transformations can be applied highly efficiently and convenient for derived quantities as well. In this article we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave-number, separately. The results to high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard Double and long Double precision arithmetic is demonstrated. Following Bessel's inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [5 × 5] arcmin resolution.
Studia Geophysica et Geodaetica, 2014
Physica A: Statistical Mechanics and its Applications, 1999
We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressure... more We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T1 and T2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particules. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter ǫ = m/M , and explicitly given up to order ǫ 2. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M = m.
Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolutio... more Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential.
Entropy, 2011
Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolutio... more Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential.
Communications in Mathematical Physics, 1971
We prove the existence of the thermodynamic limit for the pressure and show that the limit is a c... more We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential. The existence and analyticίty properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion. In the special case of Monomer-Dimer systems it is established that no phase transition is possible moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Pade approximants. Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems. * Work presented in partial fullfilment of the Ph. D. Thesis.
Communications in Mathematical Physics, 1969
The Green functions of the anisotropic Heisenberg model are studied by a method which was applied... more The Green functions of the anisotropic Heisenberg model are studied by a method which was applied previously to the reduced density matrices. Integral equations are used to prove the existence of the infinite volume limit of the Green functions, and some analyticity properties with respect to the fugacity (or magnetic field), the potentials, and the complex times.
Communications in Mathematical Physics, 1975
Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlatio... more Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlation functions in terms of the group structure associated with any lattice systems. The use of Poisson formula for abelian groups gives a simple method to obtain explicit domains where the above properties are verified.
Communications in Mathematical Physics, 1978
We investigate' Ising spin systems with general ferromagnetic, translation invariant interactions... more We investigate' Ising spin systems with general ferromagnetic, translation invariant interactions, H=-^J B σ B , J β^0. We show that the critical temperature 7] for the order parameter p i defined as the temperature below which p t >0 9 is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the "equivalent" correlation functions have the same critical exponents as T^T t from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperature Γ at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R. J. Baxter et al. concerning the equality of some critical exponents.
Physica A: Statistical Mechanics and its Applications, 2005
We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy pa... more We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy particles whose interaction is mediated by purely elastic collisions with light particles, in contact at the boundary with two heat baths with fixed temperatures T + and T −. It is shown that Fourier law is satisfied with a thermal conductivity proportional to p T (x) where T (x) is the local temperature. Entropy flux and entropy production are also investigated.
Physica A: Statistical Mechanics and its Applications, 1996
A lattice system of electrons in the presence of an external potential and under the influence of... more A lattice system of electrons in the presence of an external potential and under the influence of a magnetic field is considered. The energy of the electrons is given by the Falicov-Kimball hamiltonian, where the ions give the external potential. The problem is to find the configurations of ions and the magnetic fluxes which minimize the energy of the electrons at zero temperature. Our results, valid for large coupling constant, show that periodic configurations with periodic fluxes (possibly inhomogeneous) appear for some values of the chemical potentials.
Journal of Statistical Physics
We investigate the evolution of a system composed of N non-interacting point particles of mass m ... more We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass MO\mathcal{O} (M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by X M (t)= L[1/2–(t)] where the function is independent of M. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.
Journal of Statistical Physics, 2002
We consider the evolution of a system composed of N non-interacting point particles of mass m in ... more We consider the evolution of a system composed of N non-interacting point particles of mass m in a cylindrical container divided into two regions by a movable adiabatic wall (the adiabatic piston). We study the thermodynamic limit for the piston where the area A of the cross-section, the mass M of the piston, and the number N of particles go
Encyclopedia of Mathematical Physics, 2006
Physica A: Statistical Mechanics and its Applications, 2002
We consider the evolution of a system composed of N non-interacting point particles of mass m in ... more We consider the evolution of a system composed of N non-interacting point particles of mass m in a container divided in two regions by a movable adiabatic wall (adiabatic piston). In this talk we discuss the thermodynamic limit where the area A of the container , the number N of particles, and the mass M of the piston go to infinity keeping A M and N M fixed. We show that in this limit the motion of the piston is deterministic. Introducing simplifying assumptions we discuss the approach to equilibrium and we illustrate the results with numerical simulations. The comparison with the case of a system with finite (A, N, M) will be presented.
Communications in Mathematical Physics, 1979
We introduce the surface tension for arbitrary spin systems and study its general properties. In ... more We introduce the surface tension for arbitrary spin systems and study its general properties. In particular we show that for a large class of systems, the surface tension is zero at high temperature. We also derive a geometrical condition for the surface tension to be zero at all temperature. For discrete spin systems this condition becomes a criterion to establish the existence of a phase transition associated with surface tension. This criterion is illustrated on several examples.
Communications in Mathematical Physics, 1975
It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Ki... more It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ρ, and for the spin correlation functions σ, are essentially equivalent for all ρ, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoίf process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.
Communications in Mathematical Physics, 1977
It is shown that for any KMS-state of a classical system of noncoincident particles, the distribu... more It is shown that for any KMS-state of a classical system of noncoincident particles, the distribution functions are absolutely continuous with respect to Lebesgue measure; the equivalence between KMS states and Canonical Gibbs States is then established.
Journal of Statistical Physics, 2004
The piston problem is investigated in the case where the length of the cylinder is infinite (on b... more The piston problem is investigated in the case where the length of the cylinder is infinite (on both sides) and the ratio m/M is a very small parameter, where m is the mass of one particle of the gaz and M is the mass of the piston. Introducing initial conditions such that the stochastic motion of the piston remains in the average at the origin (no drift), it is shown that the time evolution of the fluids, analytically derived from Liouville equation, agrees with the Second Law of thermodynamics. We thus have a non equilibrium microscopical model whose evolution can be explicitly shown to obey the two laws of thermodynamics.
Journal of Geodesy, 2016
The computation of spherical harmonic series in very high resolution is known to be delicate in t... more The computation of spherical harmonic series in very high resolution is known to be delicate in terms of performance and numerical stability. A major problem is to keep results inside a numerical range of the used data type during calculations as under-/overflow arises. Extended data types are currently not desirable since the arithmetic complexity will grow exponentially with higher resolution levels. If the associated Legendre functions are computed in spectral domain then regular grid transformations can be applied highly efficiently and convenient for derived quantities as well. In this article we compare three recursive computations of the associated Legendre functions as trigonometric series, thereby ensuring a defined numerical range for each constituent wave-number, separately. The results to high degree and order show the numerical strength of the proposed method. First, the evaluation of Fourier coefficients of the associated Legendre functions has been done with respect to the floating-point precision requirements. Secondly, the numerical accuracy in the cases of standard Double and long Double precision arithmetic is demonstrated. Following Bessel's inequality the obtained accuracy estimates of the Fourier coefficients are directly transferable to the associated Legendre functions themselves and to derived functionals as well. Therefore, they can provide an essential insight to modern geodetic applications that depend on efficient spherical harmonic analysis and synthesis beyond [5 × 5] arcmin resolution.
Studia Geophysica et Geodaetica, 2014
Physica A: Statistical Mechanics and its Applications, 1999
We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressure... more We consider a one-dimensional system consisting of two infinite ideal fluids, with equal pressures but different temperatures T1 and T2, separated by an adiabatic movable piston whose mass M is much larger than the mass m of the fluid particules. This is the infinite version of the controversial adiabatic piston problem. The stationary non-equilibrium solution of the Boltzmann equation for the velocity distribution of the piston is expressed in powers of the small parameter ǫ = m/M , and explicitly given up to order ǫ 2. In particular it implies that although the pressures are equal on both sides of the piston, the temperature difference induces a non-zero average velocity of the piston in the direction of the higher temperature region. It thus shows that the asymmetry of the fluctuations induces a macroscopic motion despite the absence of any macroscopic force. This same conclusion was previously obtained for the non-physical situation where M = m.
Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolutio... more Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential.
Entropy, 2011
Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolutio... more Following the analytic approach to thermodynamics developed by Stückelberg, we study the evolution equations of a closed thermodynamic system consisting of point particles in a fluid. We obtain a system of coupled differential equations describing the mechanical and the thermal evolution of the system. The coupling between these evolution equations is due to the action of a viscous friction term. Finally, we apply our coupled evolution equations to study the thermodynamics of an isolated system consisting of identical point particles interacting through a harmonic potential.
Communications in Mathematical Physics, 1971
We prove the existence of the thermodynamic limit for the pressure and show that the limit is a c... more We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential. The existence and analyticίty properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion. In the special case of Monomer-Dimer systems it is established that no phase transition is possible moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Pade approximants. Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems. * Work presented in partial fullfilment of the Ph. D. Thesis.
Communications in Mathematical Physics, 1969
The Green functions of the anisotropic Heisenberg model are studied by a method which was applied... more The Green functions of the anisotropic Heisenberg model are studied by a method which was applied previously to the reduced density matrices. Integral equations are used to prove the existence of the infinite volume limit of the Green functions, and some analyticity properties with respect to the fugacity (or magnetic field), the potentials, and the complex times.
Communications in Mathematical Physics, 1975
Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlatio... more Asano-Ruelle-Slawny method is generalized to discuss analyticity and uniqueness of the correlation functions in terms of the group structure associated with any lattice systems. The use of Poisson formula for abelian groups gives a simple method to obtain explicit domains where the above properties are verified.
Communications in Mathematical Physics, 1978
We investigate' Ising spin systems with general ferromagnetic, translation invariant interactions... more We investigate' Ising spin systems with general ferromagnetic, translation invariant interactions, H=-^J B σ B , J β^0. We show that the critical temperature 7] for the order parameter p i defined as the temperature below which p t >0 9 is independent of the way in which the symmetry breaking interactions approach zero from above. Furthermore, all the "equivalent" correlation functions have the same critical exponents as T^T t from below, e.g. for pair interactions all the odd correlations have the same critical index as the spontaneous magnetization. The number of fluid and crystalline phases (periodic equilibrium states) coexisting at a temperature Γ at which the energy is continuous is shown to be related to the number of symmetries of the interactions. This generalizes previous results for Ising spins with even (and non-vanishing nearest-neighbour) ferromagnetic interactions. We discuss some applications of these results to the triangular lattice with three body interactions and to the Ashkin-Teller model. Our results give the answer to the question raised by R. J. Baxter et al. concerning the equality of some critical exponents.
Physica A: Statistical Mechanics and its Applications, 2005
We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy pa... more We investigate the stationary nonequilibrium states of a quasi one-dimensional system of heavy particles whose interaction is mediated by purely elastic collisions with light particles, in contact at the boundary with two heat baths with fixed temperatures T + and T −. It is shown that Fourier law is satisfied with a thermal conductivity proportional to p T (x) where T (x) is the local temperature. Entropy flux and entropy production are also investigated.