Abstract
Abstract
The large deviations at various levels that are explicit for Markov jump processes satisfying detailed balance are revisited in terms of the supersymmetric quantum Hamiltonian H that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density
p
^
(
C
)
of the configurations C seen during a trajectory over the large time window
[
0
,
T
]
, and rewrite the explicit Donsker–Varadhan rate function as the matrix element
I
[
2
]
[
p
^
(
.
)
]
=
⟨
p
^
|
H
|
p
^
⟩
involving the square-root ket
|
p
^
⟩
. (The analog formula is also discussed for reversible diffusion processes as a comparison.) We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density
p
^
(
C
)
and the empirical local activities
a
^
(
C
,
C
′
)
characterizing the density of jumps between two configurations
(
C
,
C
′
)
. Finally, the explicit rate function for the joint probability of the empirical density
p
^
(
C
)
and of the empirical total activity
A
^
that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements
⟨
p
^
|
H
|
p
^
⟩
and
⟨
p
^
|
H
off
|
p
^
⟩
, where
H
off
represents the off-diagonal part of the supersymmetric Hamiltonian H. This general framework is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian H corresponds to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites. It is then useful to introduce the quantum density matrix
ρ
^
=
|
p
^
⟩
⟨
p
^
|
associated with the empirical density
p
^
(
.
)
in order to rewrite the various rate functions in terms of reduced density matrices involving only two or three neighboring sites.