Duality and hidden equilibrium in transport models

We thank the referees for their interesting reports. We modified our manuscript accordingly. Please find the changes below.

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Referee 1:

Questions:

(1) The full equivalence of trajectories at all times (not just long times) requires the time-dependent form of the transformation. Could the authors please comment on this.

(2) Specifically, the initial probability in the original problem (generated by H) gets mapped to a different probability in the transformed problem (generated by H_eq) by an unpaired gauge operator P. How do these probabilities relate to each other? Furthermore, for the overall probability over trajectories to be well behaved the usual condition is that the transformation reverts eventually to the identity at the final time, see e.g. [3]. Does not the same need to occur here?

(3) If one is interested in steady state dynamics, the initial state becomes irrelevant. However, in light of the intriguing comment by the authors towards the end about the free-energy of H_eq problem being the non-equilibrium free-energy of the driven problem there is a related consideration. To obtain the probability over states one needs to sandwich (using the MPS language) the train of operators with a configuration vector. But again there is another unpaired transformation operator that maps one basis to the other. I think one then needs to explain more clearly how the steady state probabilities in one problem map to the steady state probabilities in the other. I do not think it is just p(x) = p_eq(x), and thus it is not totally obvious to me what the statement about the free-energy precisely means. Can the authors please comment?

I apologise if some or all of these are already answered in the MS, but in that case I would suggest making those statements more prominent and explicit.

[1] Haegeman J, Cirac J I, Osborne T J and Verstraete F 2013 Phys. Rev. B 88 085118
[2] Chetrite R and Gupta S 2011 J. Stat. Phys. 143 543
[3] Chetrite R and Touchette H 2015 Ann. Henri Poincaré 16 2005

Answers/modifications:

(1) The mapping in eq. (25) of the revised version is a similarity transformation
between the (transposed) generators of the equilibrium process and of the
non-equilibrium process. As it is now explained in the paragraph
following eq. (15), the mapping provides then the possibility, AT ALL TIMES,
of expressing expectations of the non-equilibrium process via expectations of a
dual absorbing process. Even though the use of the dual process becomes particularly
convenient in the long-time limit where the dual system will be void,
the fact remains that any process in the original system is mappable
to a process either with absorbing baths or with equilibrium baths.
This is interesting as a matter of principle.
We did not investigate further the relation between the paths-space
measures of equilibrium process and of the non-equilibrium process for a given
fixed time.

(2) The relation between the probability distribution of the original process
and the probability distribution of the transformed process is now explicit
in the text that has been added after eq. (26). See also the comment at the end of section 4.

(3) For the mapping between the steady state probabilities in one problem and the steady state probabilities in the other, see the answers above. For the statement about free energies:
this indeed deserves further investigation. While the statement is clear (being the
equilibrium free energy well defined), computing the free energy of the
non-equilibrium system via the mapping to equilibrium requires non trivial work.
This could possibly be done in the integrable case of the open symmetric exclusion process
and should reproduce the density large deviation functionals that has been found
in the macroscopic fluctuation theory of Bertini et al. We plan to reconsider this
problem in a future paper. In any case, for the use of the reader we have added the three references dealing with MPS language.

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Referee 2:

Questions:
(1) In case of the integrable symmetric exclusion process the similarity
transformation has been explicitly constructed by one of the authors
in a recent work. While this is mentioned above (4.23) it is left unclear
what parts of the following discussion is new, and what is quoted from
said earlier work. I think the authors should be more explicit about
this. As far as I can tell (4.31) is old, but (4.36) is new.

(2) The authors may want to add a reference to the work by Alcaraz et
al mentioned above to establishing the isospectrality of
"Hamiltonians" for stochastic processes differing by operators that
are upper triangular in the right basis, as is the case for (4.21) and
(4.22).

(3) My main question has to do with the authors' statement that their
class of transport models are "in fact hidden equilibrium models". My
understanding is that the authors have established that the evolution
operators of the processes they study can be mapped, by means of a
similarity transformation, to "Hamiltonians" that look like they
describe equilibrium situations. I am confused about the authors
statement quoted above because these "Hamiltonians" are generally not stochastic
and therefore cannot be thought of as generators of stochastic
processes. The steady state of the original stochastic process can of
course by construction be obtained from the "ground state" of the
transformed "Hamiltonian" (now viewed as a quantum spin chain at zero
temperature), but this looks like a purely technical observation to me
(in the sense that expectation values in the quantum model will not
generally be related to averages in the original stochastic
process). I think it would be very helpful if the authors expressed
more precisely in which sense their class of transport models are "in
fact hidden equilibrium models".

Answers/modifications:

(1) We have extended the discussion about the relation to the work
of one of the authors and the current article just above (37).

(2) We now refer to the work of Alcaraz in the text preceding eq. (21).

(3) The two Hamiltonian appearing in eq. (25) are both stochastic,
H is the evolution operator of the non-equilibrium process and H_eq
is the evolution operator of the equilibrium process. We added
some text before eq. (27) to make explicit the relation between
the probability distributions of the two processes.
See also the comment at the end of section 4, where it is discussed the
situation in which the initial state is not a probability measure.

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Referee 3:

Questions:
(1) A general notion of duality is mentioned but never introduced in a general setting, apart from stating that it is a well-known duality in probability theory. Also, it is not clear how the existence of a symmetry group for the generator interplays with such duality. Is there a general way to introduce group-based diffusive models and duality independently of the specific cases analyzed?

(2) The technical knowledge displayed of the rich mathematics of interacting particle models is impressive, but at times it looks like a piece of virtuosism (for example: what is the actual need for eq. 4.9? Also: in Sec: 4.3 it is mentioned that, quite intriguingly, integrability allows to solve for W whenever Q_0 is easier to diagonalize than H_0: but it appears that in the following treatment an expression for W is found without diagonalizing H_0)

(3) In the introduction it is mentioned that there exist a time-reversed mapping between \rho_2 and \rho_1 even when the system is driven out of equilibrium. But how is this special of equilibrium systems? Given a MC propagator e^tH one can always define the inverse map e^-tH. Detailed balance has more to do with correlation functions than with probabilities.

(4) It is mentioned in passing that the new detailed balanced dynamics has absorbing sites. But then, the stationary state of the new equilibrium dynamics will be different from the stationary state of the original dynamics. So, while there is a mapping between generators, it appears that there is no mapping between the time-evolved probabilities, and that stationary states map to specific transient states. Maybe the Authors should explain better what exactly the mapping consists of.

Answers/modifications:

(1) We added a paragraph after eq. (15) where we show how our formulation is essentially equivalent
to the notion of duality. We pointed to references discussing the relation between duality
and symmetries, as well as a constructive approach to group-based models with duality. However the referee is right that a complete theory of duality
is still missing.

(2) For clarity we pointed to the formal aspect of formula (4.9). Further, we reformulated the sentence in section 4.3 "Q_0 is easier to diagonalise" in order to clarify that we do not need the eigenvectors of the unperturbed system but the eigenvalues to obtain the similarity transformation W.

(3) Unfortunately e^{-tH} is not a stochastic process.

(4) The precise implication of the mapping is now detailed for the probability vector.
See the added paragraph after eq. (27).

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