We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted $P_{S,λ,t}$ , concentrated on the set of energy differences $σ(H_S)-σ(H_S)$ ( $σ(H_S)$ is the spectrum of the Hamiltonian of S, $t$ is the length of the time interval during which the measurement of the energy transfer is performed, and $λ$ is the strength of the interaction between S and R). The second FCS, $P_{R,λ,t}$ , describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $t\rightarrow\infty$ of these two measures followed by the weak coupling limit $λ\rightarrow 0$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $P_{R,λ,t}$ by quantum transfer operators.We study a microscopic Hamiltonian model describing an N-level quantum system S coupled to an infinitely extended thermal reservoir R. Initially, the system S is in an arbitrary state while the reservoir is in thermal equilibrium at temperature T. Assuming that the coupled system S+R is mixing with respect to the joint thermal equilibrium state, we study the Full Counting Statistics (FCS) of the energy transfers S->R and R->S in the process of return to equilibrium. The first FCS describes the increase of the energy of the system S. It is an atomic probability measure, denoted $P_{S,λ,t}$, concentrated on the set of energy differences $σ(H_S)-σ(H_S)$ ($σ(H_S)$ is the spectrum of the Hamiltonian of S, $t$ is the length of the time interval during which the measurement of the energy transfer is performed, and $λ$ is the strength of the interaction between S and R). The second FCS, $P_{R,λ,t}$, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit $t\rightarrow\infty$ of these two measures followed by the weak coupling limit $λ\rightarrow 0$ and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of $P_{R,λ,t}$ by quantum transfer operators.

arXiv:1409.8610Sep 30, 20140 citations

Another Return of “Return to Equilibrium”

Jürg Martin Fröhlich, Marco Merkli

arXiv:math-ph/0410011Communications in Mathematical PhysicsSep 17, 200451 citations

Energy Full Counting Statistics and Return to Equilibrium

Jane Panangaden

We consider a finite dimensional quantum system $S$ in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir $R$ in equilibrium at inverse temperature $β$ . The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter $λ$ . We assume the system $S + R$ has the property of return to equilibrium, which means that after sufficiently long time, the joint system will have reached equilibrium at inverse temperature $β$ . In this context, we prove a refinement to the first law of thermodynamics, which states that the total energy of the system and reservoir is conserved. Specifically, we define two measures which encode all the information about the fluctuations of the system and reservoir energy when two measurements are made at time $0$ and time $t$ . These measures are called the full counting statistics (FCS). We prove weak convergence of the system and reservoir FCS in the double limit $t \rightarrow \infty$ and $λ\rightarrow 0$ .We consider a finite dimensional quantum system $S$ in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir $R$ in equilibrium at inverse temperature $β$. The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter $λ$. We assume the system $S + R$ has the property of return to equilibrium, which means that after sufficiently long time, the joint system will have reached equilibrium at inverse temperature $β$. In this context, we prove a refinement to the first law of thermodynamics, which states that the total energy of the system and reservoir is conserved. Specifically, we define two measures which encode all the information about the fluctuations of the system and reservoir energy when two measurements are made at time $0$ and time $t$. These measures are called the full counting statistics (FCS). We prove weak convergence of the system and reservoir FCS in the double limit $t \rightarrow \infty$ and $λ\rightarrow 0$.

arXiv:2110.01678Oct 4, 20210 citations

Another Return of “Return to Equilibrium”

Jürg Martin Fröhlich, Marco Merkli

Nov 1, 200444 citations

Lindblad equation for a non-interacting fermionic system: full-counting statistics

Mariya V. Medvedyeva, Stefan K. Kehrein

We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. By the basis change the Liouvillian operator is brought to the quadratic form. This allows us a straightforward calculation of any observable in the non-equilibrium steady state. We introduce the counting field in the Lindblad equation which brings us to the generating function and helps us to obtain all cumulants of the charge transport. For the two-site system we give the expression for the generating function. For system longer than two sites we perform numerical investigations which suggest that it in a uniform system the cumulants of order $k$ are independent of the size of the system for system sizes larger $k+1$ . The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.We develop a method of calculating the full-counting statistics for a non-interacting fermionic system coupled to the memory-less reservoirs. The evolution of the system is described by the Lindblad equation. By the basis change the Liouvillian operator is brought to the quadratic form. This allows us a straightforward calculation of any observable in the non-equilibrium steady state. We introduce the counting field in the Lindblad equation which brings us to the generating function and helps us to obtain all cumulants of the charge transport. For the two-site system we give the expression for the generating function. For system longer than two sites we perform numerical investigations which suggest that it in a uniform system the cumulants of order $k$ are independent of the size of the system for system sizes larger $k+1$. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.

arXiv:1310.4997Oct 18, 20134 citations

P_{S,λ,t}

, concentrated on the set of energy differences

σ(H_S)-σ(H_S)

(

σ(H_S)

is the spectrum of the Hamiltonian of S,

t

is the length of the time interval during which the measurement of the energy transfer is performed, and

λ

is the strength of the interaction between S and R). The second FCS,

P_{R,λ,t}

, describes the decrease of the energy of the reservoir R and is typically a continuous probability measure whose support is the whole real line. We study the large time limit

t\rightarrow\infty

of these two measures followed by the weak coupling limit

λ\rightarrow 0

and prove that the limiting measures coincide. This result strengthens the first law of thermodynamics for open quantum systems. The proofs are based on modular theory of operator algebras and on a representation of

P_{R,λ,t}

by quantum transfer operators.

We consider a finite dimensional quantum system

S

in an arbitrary initial state coupled to an infinitely extended quantum thermal reservoir

R

in equilibrium at inverse temperature

β

. The coupling is given by a bounded perturbation of the dynamics and the coupling strength is controlled by a parameter

λ

. We assume the system

S + R

has the property of return to equilibrium, which means that after sufficiently long time, the joint system will have reached equilibrium at inverse temperature

β

. In this context, we prove a refinement to the first law of thermodynamics, which states that the total energy of the system and reservoir is conserved. Specifically, we define two measures which encode all the information about the fluctuations of the system and reservoir energy when two measurements are made at time

0

and time

t

. These measures are called the full counting statistics (FCS). We prove weak convergence of the system and reservoir FCS in the double limit

t \rightarrow \infty

and

λ\rightarrow 0

k

are independent of the size of the system for system sizes larger

k+1

. The counting statistics from the Lindblad approach does not take into account interference in the reservoirs which gives a decreased noise in comparison with the Green function method which describes phase coherent leads. The current obtained by two methods is the same, which relies on the current conservation. The Fano factors are different (with a linear relation connecting them) and allow to distinguish between memory-less and phase coherent reservoirs.