Seminario de Teoría Espectral
2018-06-21
17:00hrs.
17:00hrs.
Marouane Assal. Facultad de Matemáticas, PUC
Semiclassical Trace Formula and Spectral Shift Function for Schrödinger Operators With Matrix-Valued Potentials
Sala 1
http://www.mat.uc.cl/~graikov/seminar.html
Semiclassical Trace Formula and Spectral Shift Function for Schrödinger Operators With Matrix-Valued Potentials
Sala 1
http://www.mat.uc.cl/~graikov/seminar.html
2018-05-24
17:00hrs.
17:00hrs.
Christian Sadel. Facultad de Matemáticas, UC
One-Channel Operators, a General Radial Transfer Matrix Approach and Absolutely Continuous Spectrum
sala 1
Abstract:
First I will introduce one-channel operators and their spectral theory analyses through transfer matrices solving the eigenvalue equation.
One-Channel Operators, a General Radial Transfer Matrix Approach and Absolutely Continuous Spectrum
sala 1
Abstract:
First I will introduce one-channel operators and their spectral theory analyses through transfer matrices solving the eigenvalue equation.
Then, inspired from the specific form of these transfer matrices, we will define sets of transfer matrices for any discrete Hermitian operator with locally finite hopping by considering quasi-spherical partitions.
A generalization of some spectral averaging formula for Jacob operators is given and criteria for the existence and pureness of absolutely continuous spectrum are derived.
In the one-channel case this already led to several examples of existence of absolutely continuous spectrum for the Anderson models on such graphs with finite dimensional growth (of dimension $d>2$).
The method has some potential of attacking the open extended states conjecture for the Anderson model in $\mathbb{Z}^d, d\geq 3$.
2018-05-17
17:00hrs.
17:00hrs.
Humberto Prado. Universidad de Santiago de Chile
The Spectral Theorem in The Study of The Fractional Schrödinger Equation
Sala 1
Abstract:
We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family. Examples will be discussed.
http://www.mat.uc.cl/~graikov/seminar.html
The Spectral Theorem in The Study of The Fractional Schrödinger Equation
Sala 1
Abstract:
We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family. Examples will be discussed.
http://www.mat.uc.cl/~graikov/seminar.html
2018-05-03
17:00hrs.
17:00hrs.
Georgi Raikov. Pontificia Universidad Católica de Chile
Lifshits Tails for Randomly Twisted Quantum Waveguides
Sala 1
Abstract:
I will consider the Dirichlet Laplacian on a three-dimensional twisted waveguide with random Anderson-type twisting. I will discuss the Lifshits tails for the related integrated density of states (IDS), i.e. the asymptotics of the IDS as the energy approaches from above the infimum of its support. In particular, I will specify the dependence of the Lifshits exponent on the decay rate of the single-site twisting.
The talk is based on joint works with Werner Kirsch (Hagen) and David Krejcirik (Prague).
Lifshits Tails for Randomly Twisted Quantum Waveguides
Sala 1
Abstract:
I will consider the Dirichlet Laplacian on a three-dimensional twisted waveguide with random Anderson-type twisting. I will discuss the Lifshits tails for the related integrated density of states (IDS), i.e. the asymptotics of the IDS as the energy approaches from above the infimum of its support. In particular, I will specify the dependence of the Lifshits exponent on the decay rate of the single-site twisting.
The talk is based on joint works with Werner Kirsch (Hagen) and David Krejcirik (Prague).
2018-03-27
17:00hrs.
17:00hrs.
Rajinder Mavi. Michigan State University
Anderson Localization for a Disordered Polaron
Sala 2, Facultad de Matemáticas
Abstract:
We will consider an operator modeling a tracer particle on the integer lattice subject to an Anderson field, we associate a one dimensional oscillator to each site of the lattice. This forms a polaron model where the oscillators communicate only through the hopping of the tracer particle. This introduces, a priori, infinite degeneracies of bare energies at large distances. We nevertheless show Dynamical Localization of the tracer particle for compact subsets of the spectrum.
This is joint work with Jeff Schenker.
Anderson Localization for a Disordered Polaron
Sala 2, Facultad de Matemáticas
Abstract:
We will consider an operator modeling a tracer particle on the integer lattice subject to an Anderson field, we associate a one dimensional oscillator to each site of the lattice. This forms a polaron model where the oscillators communicate only through the hopping of the tracer particle. This introduces, a priori, infinite degeneracies of bare energies at large distances. We nevertheless show Dynamical Localization of the tracer particle for compact subsets of the spectrum.
This is joint work with Jeff Schenker.
2018-03-22
17:00hrs.
17:00hrs.
Timo Weidl . Universität Stuttgart
Sharp Semiclassical Estimates With Remainder Terms
Sala 1
Abstract:
Sharp semi-classical spectral estimates give uniform bounds on eigenvalue sums in terms of their Weyl asymptotics. Famous examples are the Li-Yau and the Berezin inequalities on eigenvalues of the Dirichlet Laplacian in domains. Recently these bounds have been sharpened with additional remainder terms, as in the Melas inequality. I give an overview on some of these results and, in particular, I will talk on a Melas type bound for the two-dimensional Dirichlet Hamiltonian with constant magnetic field in a bounded domain.
Sharp Semiclassical Estimates With Remainder Terms
Sala 1
Abstract:
Sharp semi-classical spectral estimates give uniform bounds on eigenvalue sums in terms of their Weyl asymptotics. Famous examples are the Li-Yau and the Berezin inequalities on eigenvalues of the Dirichlet Laplacian in domains. Recently these bounds have been sharpened with additional remainder terms, as in the Melas inequality. I give an overview on some of these results and, in particular, I will talk on a Melas type bound for the two-dimensional Dirichlet Hamiltonian with constant magnetic field in a bounded domain.
2017-11-30
17:00hrs.
17:00hrs.
Diomba Sambou. Facultad de Matemáticas, PUC
On The Discrete Spectrum of Non-Self-Adjoint Pauli Operators With Non Constant Magnetic Fields
Sala 1
Abstract:
http://www.mat.uc.cl/~graikov/seminar.html
On The Discrete Spectrum of Non-Self-Adjoint Pauli Operators With Non Constant Magnetic Fields
Sala 1
Abstract:
I will talk about the discrete spectrum generated by complex matrix-valued perturbations for a class of 2D and 3D Pauli operators with non-constant admissible magnetic fields. We shall establish a simple criterion for the potentials to produce discrete spectrum near the low ground energy of the operators. Moreover, in case of creation of non-real eigenvalues, this criterion specifies also their location.
http://www.mat.uc.cl/~graikov/seminar.html
2017-11-23
17:00hrs.
17:00hrs.
Pablo Miranda. Universidad de Santiago de Chile
Resonancias en Guías de Ondas Torcidas
Sala 1
Abstract:
En esta charla consideraremos el Laplaciano definido en una guía de ondas recta, la cual será torcida localmente. Se sabe que tal perturbación no crea valores propios discretos. Sin embargo, es posible definir una extensión meromorfa de la resolvente del Laplaciano perturbado, la que nos permite mostrar que existe exactamente una resonancia cerca de ínfimo del espectro esencial. Para esta resonancia calcularemos su comportamiento asintótico, en función del tamaño del torcimiento. Por último daremos una idea de como extender estos resultados para los "umbrales" superiores en el espectro del Laplaciano no perturbado.
Resonancias en Guías de Ondas Torcidas
Sala 1
Abstract:
En esta charla consideraremos el Laplaciano definido en una guía de ondas recta, la cual será torcida localmente. Se sabe que tal perturbación no crea valores propios discretos. Sin embargo, es posible definir una extensión meromorfa de la resolvente del Laplaciano perturbado, la que nos permite mostrar que existe exactamente una resonancia cerca de ínfimo del espectro esencial. Para esta resonancia calcularemos su comportamiento asintótico, en función del tamaño del torcimiento. Por último daremos una idea de como extender estos resultados para los "umbrales" superiores en el espectro del Laplaciano no perturbado.
2017-11-16
17:00hrs.
17:00hrs.
Siegfried Beckus. Technion, Haifa, Israel
Shnol Type Theorem for The Agmon Ground State
Sala 1
Abstract:
http://www.mat.uc.cl/~graikov/seminar.html
Shnol Type Theorem for The Agmon Ground State
Sala 1
Abstract:
The celebrated Shnol theorem asserts that every polynomially bounded generalized eigenfunction for a given energy E associated with a Schrodinger operator H implies that E is in the L2-spectrum of H. Later Simon rediscorvered this result independently and proved additionally that the set of energies admiting a polynomially bounded generalized eigenfunction is dense in the spectrum. A remarkable extension of these results hold also in the Dirichlet setting. It was conjectured that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During the talk, we positively answer the conjecture indicating that the Agmon ground state describes the spectrum of the operator H. Specifically, we show that if u is a generalized eigenfunction for the eigenvalue E that is bounded by the Agmon ground state then E belongs to the L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a suitable notion of Agmon ground state is available.
http://www.mat.uc.cl/~graikov/seminar.html
2017-11-09
17:00hrs.
17:00hrs.
Vincent Bruneau. Université de Bordeaux, France
Spectral Analysis in The Large Coupling Limit for Singular Perturbations
Sala 1
Abstract:
We consider a singular perturbation of the Laplacian, supported on a bounded domain with a large coupling constant.
We study the asymptotic behavior of spectral quantities (eigenvalues and resonances) when the coupling constant tends to infinity.
Joint work with G. Carbou.
http://www.mat.uc.cl/~graikov/seminar.html
Spectral Analysis in The Large Coupling Limit for Singular Perturbations
Sala 1
Abstract:
We consider a singular perturbation of the Laplacian, supported on a bounded domain with a large coupling constant.
We study the asymptotic behavior of spectral quantities (eigenvalues and resonances) when the coupling constant tends to infinity.
Joint work with G. Carbou.
http://www.mat.uc.cl/~graikov/seminar.html
2017-10-26
17:00hrs.
17:00hrs.
Dr. Guo Chuan Thiang . University of Adelaide
Time-Reversal, Monopoles, and Equivariant Topological Matter
Sala 1
Abstract:
A crucial feature of experimentally discovered topological insulators (2008) and semimetals (2015) is time-reversal, which realises an order-two symmetry "Quaternionically''. Guided by physical intuition, I will formulate a certain equivariant Poincare duality which allows a useful visualisation of "Quaternionic'' characteristic classes and the concept of Euler structures. I also identify a new monopole with torsion charge, and show how the experimental signature of surface Fermi arcs are holographic versions of bulk Dirac strings.
http://www.mat.uc.cl/~graikov/seminar.html
Time-Reversal, Monopoles, and Equivariant Topological Matter
Sala 1
Abstract:
A crucial feature of experimentally discovered topological insulators (2008) and semimetals (2015) is time-reversal, which realises an order-two symmetry "Quaternionically''. Guided by physical intuition, I will formulate a certain equivariant Poincare duality which allows a useful visualisation of "Quaternionic'' characteristic classes and the concept of Euler structures. I also identify a new monopole with torsion charge, and show how the experimental signature of surface Fermi arcs are holographic versions of bulk Dirac strings.
http://www.mat.uc.cl/~graikov/seminar.html
2017-09-28
17:00hrs.
17:00hrs.
Georgi Raikov. Facultad de Mateméticas, PUC
Teoría Espectral de Hamiltonianos Cuánticos. II
Sala 1
http://www.mat.uc.cl/~graikov/seminar.html
Teoría Espectral de Hamiltonianos Cuánticos. II
Sala 1
http://www.mat.uc.cl/~graikov/seminar.html
2017-09-21
17:00hrs.
17:00hrs.
Georgi Raikov. Facultad de Matemáticas, PUC
Teoría Espectral de Hamiltonianos Cuánticos. I
Sala 1
Abstract:
En la segunda charla se hablará de temas más avanzados como hamiltonianos fibrados (por ejemplo, operadores con coeficientes periódicos) y hamiltonianos ergódicos.
Como ya indicado, el ciclo está orientado hacia alumnos de licenciatura pero puede ser interesante también para alumnos de postgrado y postdoctorados.
Teoría Espectral de Hamiltonianos Cuánticos. I
Sala 1
Abstract:
El propósito principal del ciclo de dos charlas "Teoría espectral de hamiltonianos cuánticos" del 21 y 28 de septiembre es despertar el interés de alumnos de licenciatura hacia los problemas matemáticos que aparecen en la teoría espectral de hamiltonianos cuánticos (operadores auto-adjuntos en espacios de Hilbert).
En la primera charla se presentarán algunos hechos básicos como el teorema espectral, la idea de cuantificación y las propiedades básicas de los operadores de Schrödinger, Pauli y Dirac.
2017-08-10
17:00hrs.
17:00hrs.
Soeren Fournais. Aarhus Universiy
A Hardy-Lieb-Thirring Inequality for Fractional Pauli Operators
Sala 1
Abstract:
In this talk we will discuss recent work on Hardy-Lieb-Thirring inequalities for the Pauli operator. The classical Lieb-Thirring inequality estimates the sum of the negative eigenvalues of a Schrödinger operator
$-\Delta + V$ by an integral of a power of the potential. In $3$-dimensions, this becomes
$$
\operatorname{tr}( - \Delta + V)_{-} \leq C \int (V(x))_{-}^{5/2}\,dx
$$
The classical Hardy inequality states that (also in $3D$),
$$
-\Delta - \frac{1}{4 |x|^2} \geq 0,
$$
where the constant $\frac{1}{4}$ is the sharp constant for this bound.
It is well known, that these inequalities can be combined to yield "Hardy-Lieb-Thirring inequalities”, i.e. the Lieb-Thirring inequality above still holds (possibly with a different constant) if $V$ is replaced by $- \frac{1}{4 |x|^2} + V$ on the left side.
In this talk we will discuss similar inequalities, where the non-relativistic kinetic energy operator $-\Delta$ is replaced by a magnetic Pauli operator. In particular, we will discuss a relativistic version, where the kinetic energy is the square root of a Pauli operator, and where $ \frac{1}{4 |x|^2}$ is replaced by $\frac{c_H}{|x|}$, with $c_H$ being the critical Hardy constant for the relativistic problem.
This is joint work with Gonzalo Bley.
A Hardy-Lieb-Thirring Inequality for Fractional Pauli Operators
Sala 1
Abstract:
In this talk we will discuss recent work on Hardy-Lieb-Thirring inequalities for the Pauli operator. The classical Lieb-Thirring inequality estimates the sum of the negative eigenvalues of a Schrödinger operator
$-\Delta + V$ by an integral of a power of the potential. In $3$-dimensions, this becomes
$$
\operatorname{tr}( - \Delta + V)_{-} \leq C \int (V(x))_{-}^{5/2}\,dx
$$
The classical Hardy inequality states that (also in $3D$),
$$
-\Delta - \frac{1}{4 |x|^2} \geq 0,
$$
where the constant $\frac{1}{4}$ is the sharp constant for this bound.
It is well known, that these inequalities can be combined to yield "Hardy-Lieb-Thirring inequalities”, i.e. the Lieb-Thirring inequality above still holds (possibly with a different constant) if $V$ is replaced by $- \frac{1}{4 |x|^2} + V$ on the left side.
In this talk we will discuss similar inequalities, where the non-relativistic kinetic energy operator $-\Delta$ is replaced by a magnetic Pauli operator. In particular, we will discuss a relativistic version, where the kinetic energy is the square root of a Pauli operator, and where $ \frac{1}{4 |x|^2}$ is replaced by $\frac{c_H}{|x|}$, with $c_H$ being the critical Hardy constant for the relativistic problem.
This is joint work with Gonzalo Bley.
2017-05-18
17:00hrs.
17:00hrs.
Silvius Klein . PUC Rio de Janeiro
Anderson Localization for One-Frequency Quasi-Periodic Block Jacobi Operators
Sala 1
Abstract:
Consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. This model is a natural generalization of Schroedinger operators of this kind. It contains all finite range hopping Schroedinger operators on integer or band integer lattices.
In this talk I will discuss a recent result concerning Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency.
Anderson Localization for One-Frequency Quasi-Periodic Block Jacobi Operators
Sala 1
Abstract:
Consider a one-frequency, quasi-periodic, block Jacobi operator, whose blocks are generic matrix-valued analytic functions. This model is a natural generalization of Schroedinger operators of this kind. It contains all finite range hopping Schroedinger operators on integer or band integer lattices.
In this talk I will discuss a recent result concerning Anderson localization for this type of operator under the assumption that the coupling constant is large enough but independent of the frequency.
2017-04-06
17:00hrs.
17:00hrs.
Sébastien Breteaux . Basque Center for Applied Mathematics
The Time-Dependent Hartree-Fock-Bogoliubov Equations for Bosons
Sala 1
Abstract:
Joint work with V. Bach, T. Chen, J. Fröhlich, and I. M. Sigal.
It was first predicted in 1925 by Einstein (generalizing a previous work of Bose) that, at very low temperatures, identical Bosons could occupy the same state. This large assembly of Bosons would then form a quantum state of the matter which could be observed at the macroscopic scale. The first experimental realisation of a gas condensate was then done in 1995 by Cornell and Wieman, and this motivated numerous works on Bose-Einstein condensation.
In particular, we are interested in the dynamics of such a condensate. To describe the dynamics of such a condensate, the first approximation is the time dependent Gross-Pitaevskii equation, or, in an other scaling, the Hartree equation. To precise this description, we derive the time-dependent Hartree-Fock-Bogoliubov equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate via quasifree reduction. We prove global well posedness for the HFB equations for sufficiently regular interaction potentials. We show that the HFB equations have a symplectic structure and a structure similar to an Hamiltonian structure, which is sufficient to prove the conservation of the energy.
The Time-Dependent Hartree-Fock-Bogoliubov Equations for Bosons
Sala 1
Abstract:
Joint work with V. Bach, T. Chen, J. Fröhlich, and I. M. Sigal.
It was first predicted in 1925 by Einstein (generalizing a previous work of Bose) that, at very low temperatures, identical Bosons could occupy the same state. This large assembly of Bosons would then form a quantum state of the matter which could be observed at the macroscopic scale. The first experimental realisation of a gas condensate was then done in 1995 by Cornell and Wieman, and this motivated numerous works on Bose-Einstein condensation.
In particular, we are interested in the dynamics of such a condensate. To describe the dynamics of such a condensate, the first approximation is the time dependent Gross-Pitaevskii equation, or, in an other scaling, the Hartree equation. To precise this description, we derive the time-dependent Hartree-Fock-Bogoliubov equations describing the dynamics of quantum fluctuations around a Bose-Einstein condensate via quasifree reduction. We prove global well posedness for the HFB equations for sufficiently regular interaction potentials. We show that the HFB equations have a symplectic structure and a structure similar to an Hamiltonian structure, which is sufficient to prove the conservation of the energy.
2017-03-23
17:00hrs.
17:00hrs.
Rafael Tiedra de Aldecoa. Facultad de Matemáticas, PUC
Spectral Analysis of Quantum Walks With An Anisotropic Coin
Sala 1
Abstract:
We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.
This is a joint work with Serge Richard (Nagoya University) and Akito Suzuki (Shinshu University).
Spectral Analysis of Quantum Walks With An Anisotropic Coin
Sala 1
Abstract:
We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.
This is a joint work with Serge Richard (Nagoya University) and Akito Suzuki (Shinshu University).
2017-03-16
17:00hrs.
17:00hrs.
Hermann Schulz-Baldes. Universidad de Erlangen, Alemania
Finite Volume Calculation of Topological Invariants
Sala 1
Abstract:
Odd index pairings of K1-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of the so-called Bott operator. The index is also equal to the eta-invariant of the Bott operator. In presence of real symmetries, secondary $Z_2$-invariants can be obtained as the sign of the Pfaffian of the Bott operator. These results reconcile two complementary approaches to invariants in topological insulators. Joint work with Terry Loring.
Finite Volume Calculation of Topological Invariants
Sala 1
Abstract:
Odd index pairings of K1-group elements with Fredholm modules are of relevance in index theory, differential geometry and applications such as to topological insulators. For the concrete setting of operators on a Hilbert space over a lattice, it is shown how to calculate the resulting index as the signature of a suitably constructed finite-dimensional matrix, more precisely the finite volume restriction of the so-called Bott operator. The index is also equal to the eta-invariant of the Bott operator. In presence of real symmetries, secondary $Z_2$-invariants can be obtained as the sign of the Pfaffian of the Bott operator. These results reconcile two complementary approaches to invariants in topological insulators. Joint work with Terry Loring.
2017-01-05
17:00hrs.
17:00hrs.
Jake Fillman. Virginia Tech
Ballistic Propagation for Limit-Periodic Jacobi Operators
Sala 1
Abstract:
We will talk about the propagation of wave packets in a one-dimensional medium with limit-periodic background potential. If the amplitudes of the low-frequency modes of the potential decay sufficiently rapidly, then wavepackets travel ballistically in the sense that the group velocity is injective on the domain of the position operator. Since the underlying Hamiltonian has purely absolutely continuous spectrum, this answers a special case of a general question of J. Lebowitz regarding the relationship between ac spectrum and ballistic wavepacket spreading.
Ballistic Propagation for Limit-Periodic Jacobi Operators
Sala 1
Abstract:
We will talk about the propagation of wave packets in a one-dimensional medium with limit-periodic background potential. If the amplitudes of the low-frequency modes of the potential decay sufficiently rapidly, then wavepackets travel ballistically in the sense that the group velocity is injective on the domain of the position operator. Since the underlying Hamiltonian has purely absolutely continuous spectrum, this answers a special case of a general question of J. Lebowitz regarding the relationship between ac spectrum and ballistic wavepacket spreading.
2016-12-27
15:00hrs.
15:00hrs.
Michael Loss. Georgia Tech
Modeling Thermostats Using Master Equations
Sala 1
Abstract:
In this talk we discuss results for a model of randomly colliding particles interacting with a thermal bath, i.e., a thermostat. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium with inverse temperature $\beta$. The evolution propagates chaos and the one particle marginal, in the limit of large systems, satisfies an effective Boltzmann-type equation. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. It turns out that any initial distribution approaches the equilibrium distribution exponentially fast, both, in a proper function space as well as in relative entropy. Recent results concerning the approximation of thermostats by a large but finite heat reservoir will also be discussed. It turns out that in suitable norms the approximation can be shown to be uniformly in time, i.e., the error depends only on the size of the finite heat reservoir. This is joint work with Federico Bonetto, Hagop Tossounian and Ranjini Vaidyanathan.
Modeling Thermostats Using Master Equations
Sala 1
Abstract:
In this talk we discuss results for a model of randomly colliding particles interacting with a thermal bath, i.e., a thermostat. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium with inverse temperature $\beta$. The evolution propagates chaos and the one particle marginal, in the limit of large systems, satisfies an effective Boltzmann-type equation. The system admits the canonical distribution at inverse temperature $\beta$ as the unique equilibrium state. It turns out that any initial distribution approaches the equilibrium distribution exponentially fast, both, in a proper function space as well as in relative entropy. Recent results concerning the approximation of thermostats by a large but finite heat reservoir will also be discussed. It turns out that in suitable norms the approximation can be shown to be uniformly in time, i.e., the error depends only on the size of the finite heat reservoir. This is joint work with Federico Bonetto, Hagop Tossounian and Ranjini Vaidyanathan.
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