We consider a stochastic process of heat conduction where energy is redistributed along a chain between nearest neighbor sites via an improper beta distribution. Similar to the well-known Kipnis–Marchioro–Presutti (KMP) model, the finite chain is coupled at its ends with two reservoirs that break the conservation of energy when working at different temperatures. At variance with KMP, the model considered here is integrable, and one can write in a closed form the n-point correlation functions of the non-equilibrium steady state. As a consequence of the exact solution one, can directly prove that the system is in “local equilibrium,” which is described at the macro-scale by a product measure. Integrability manifests itself through the description of the model via the open Heisenberg chain with non-compact spins. The algebraic formulation of the model allows us to interpret its duality relation with a purely absorbing particle system as a change of representation.

1.
C.
Kipnis
,
C.
Marchioro
, and
E.
Presutti
, “
Heat flow in an exactly solvable model
,”
J. Stat. Phys.
27
(
1
),
65
74
(
1982
).
2.
P.
Krapivsky
and
B.
Meerson
, “
Fluctuations of current in nonstationary diffusive lattice gases
,”
Phys. Rev. E
86
(
3
),
031106
(
2012
); arXiv:1206.1842 [cond-mat.stat-mech].
3.
L.
Zarfaty
and
B.
Meerson
, “
Statistics of large currents in the Kipnis–Marchioro–Presutti model in a ring geometry
,”
J. Stat. Mech.: Theory Exp.
2016
(
3
),
033304
; arXiv:1512.02419 [cond-mat.stat-mech].
4.
E.
Bettelheim
,
N. R.
Smith
, and
B.
Meerson
, “
Inverse scattering method solves the problem of full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti model
,”
Phys. Rev. Lett.
128
(
13
),
130602
(
2022
); arXiv:2112.02474 [cond-mat.stat-mech].
5.
L.
Bertini
,
A.
De Sole
,
D.
Gabrielli
,
G.
Jona-Lasinio
, and
C.
Landim
, “
Macroscopic fluctuation theory
,”
Rev. Mod. Phys.
87
(
2
),
593
(
2015
); arXiv:1404.6466 [cond-mat.stat-mech].
6.
C. P.
Espigares
,
P. L.
Garrido
, and
P. I.
Hurtado
, “
Dynamical phase transition for current statistics in a simple driven diffusive system
,”
Phys. Rev. E
87
(
3
),
032115
(
2013
); arXiv:1212.4640 [cond-mat.stat-mech].
7.
C.
Bernardin
and
S.
Olla
, “
Fourier’s law for a microscopic model of heat conduction
,”
J. Stat. Phys.
121
(
3–4
),
271
289
(
2005
); arXiv:cond-mat/0502485 [cond-mat.stat-mech].
8.
C.
Giardinà
and
J.
Kurchan
, “
The Fourier law in a momentum-conserving chain
,”
J. Stat. Mech.: Theory Exp.
2005
(
05
),
P05009
; arXiv:cond-mat/0502485 [cond-mat.stat-mech].
9.
G.
Basile
,
C.
Bernardin
, and
S.
Olla
, “
Momentum conserving model with anomalous thermal conductivity in low dimensional systems
,”
Phys. Rev. Lett.
96
(
20
),
204303
(
2006
); arXiv:cond-mat/0509688 [cond-mat.stat-mech].
10.
G.
Basile
,
C.
Bernardin
, and
S.
Olla
, “
Thermal conductivity for a momentum conservative model
,”
Commun. Math. Phys.
287
(
1
),
67
98
(
2009
); arXiv:cond-mat/0601544 [cond-mat.stat-mech].
11.
C.
Giardinà
,
J.
Kurchan
,
F.
Redig
, and
K.
Vafayi
, “
Duality and hidden symmetries in interacting particle systems
,”
J. Stat. Phys.
135
(
1
),
25
55
(
2009
); arXiv:0810.1202 [math-ph].
12.
G.
Carinci
,
C.
Giardinà
,
C.
Giberti
, and
F.
Redig
, “
Duality for stochastic models of transport
,”
J. Stat. Phys.
152
(
4
),
657
697
(
2013
); arXiv:1212.3154 [math-ph].
13.
G.
Carinci
,
C.
Franceschini
,
R.
Frassek
,
C.
Giardinà
, and
F.
Redig
, “
Exact solution of an interacting particle systems (the ‘harmonic model’) and density large deviations
” (unpublished).
14.
M.
Capanna
,
D.
Gabrielli
, and
D.
Tsagkarpgiannis
, “
On a class of solvable stationary non-equilibrium states for mass exchange models
” (unpublished).
15.
R.
Frassek
,
C.
Giardinà
, and
J.
Kurchan
, “
Non-compact quantum spin chains as integrable stochastic particle processes
,”
J. Stat. Phys.
180
,
135
171
(
2019
); arXiv:1904.01048 [math-ph].
16.
R.
Frassek
and
C.
Giardinà
, “
Exact solution of an integrable non-equilibrium particle system
,”
J. Math. Phys.
63
,
103301
(
2022
); arXiv:2107.01720 [math-ph].
17.
F.
Redig
and
F.
Sau
, “
Generalized immediate exchange models and their symmetries
,”
Stochastic Processes Appl.
127
(
10
),
3251
3267
(
2017
); arXiv:1606.08692 [math.PR].
18.
B.
Van Ginkel
,
F.
Redig
, and
F.
Sau
, “
Duality and stationary distributions of the ‘immediate exchange model’ and its generalizations
,”
J. Stat. Phys.
163
(
1
),
92
112
(
2016
); arXiv:1508.04918 [math.PR].
19.
M.
Sasada
, “
Spectral gap for stochastic energy exchange model with nonuniformly positive rate function
,”
Ann. Probab.
43
(
4
),
1663
1711
(
2015
); arXiv:1305.4066 [math.PR].
20.
C.
Giardinà
,
J.
Kurchan
, and
F.
Redig
, “
Duality and exact correlations for a model of heat conduction
,”
J. Math. Phys.
48
(
3
),
033301
(
2007
); arXiv:cond-mat/0612198 [cond-mat].
21.
P. P.
Kulish
,
N. Y.
Reshetikhin
, and
E. K.
Sklyanin
, “
Yang-Baxter equation and representation theory: I
,”
Lett. Math. Phys.
5
,
393
403
(
1981
).
22.
E. K.
Sklyanin
, “
Boundary conditions for integrable quantum systems
,”
J. Phys. A: Math. Gen.
21
,
2375
3289
(
1988
).
23.
L. D.
Faddeev
, “
How algebraic Bethe ansatz works for integrable model
,” in
Relativistic Gravitation and Gravitational Radiation. Proceedings, School of Physics, Les Houches, France, September 26-October 6, 1995
(
Cambridge University Press
,
1996
), pp.
149
219
; arXiv:hep-th/9605187 [hep-th].
24.
S. É.
Derkachov
, “
Baxter’s Q-operator for the homogeneous XXX spin chain
,”
J. Phys. A: Math. Gen.
32
,
5299
5316
(
1999
); arXiv:solv-int/9902015 [solv-int].
25.
B. O.
Lange
and
M.
Neubert
, “
Renormalization group evolution of the B-meson light-cone distribution amplitude
,”
Phys. Rev. Lett.
91
,
102001
(
2003
); arXiv:hep-ph/0303082.
26.
V. M.
Braun
,
D. Y.
Ivanov
, and
G. P.
Korchemsky
, “
B-meson distribution amplitude in QCD
,”
Phys. Rev. D
69
,
034014
(
2004
); arXiv:hep-ph/0309330.
27.
A. V.
Belitsky
, “
Separation of variables for a flux tube with an end
,”
Nucl. Phys. B
957
,
115093
(
2020
); arXiv:1902.08596 [hep-th].
28.
G.
Carinci
,
C.
Giardinà
,
C.
Giberti
, and
F.
Redig
, “
Dualities in population genetics: A fresh look with new dualities
,”
Stochastic Processes Appl.
125
(
3
),
941
969
(
2015
); arXiv:1302.3206 [math.PR].
29.
C.
Franceschini
,
C.
Giardinà
, and
W.
Groenevelt
, “
Self-duality of Markov processes and intertwining functions
,”
Math. Phys., Anal. Geom.
21
(
4
),
29
(
2018
); arXiv:1801.09433 [math.PR].
30.
C.
Franceschini
,
P.
Gonçalves
, and
F.
Sau
, “
Symmetric inclusion process with slow boundary: Hydrodynamics and hydrostatics
,”
Bernoulli
28
(
2
),
1340
1381
(
2022
); arXiv:2007.11998 [math.PR].
31.
G.
Carinci
,
C.
Franceschini
,
C.
Giardinà
,
W.
Groenevelt
, and
F.
Redig
, “
Orthogonal dualities of Markov processes and unitary symmetries
,”
SIGMA
15
,
053
(
2019
); arXiv:1812.08553 [math.PR].
32.
S.
Floreani
,
F.
Redig
, and
F.
Sau
, “
Orthogonal polynomial duality of boundary driven particle systems and non-equilibrium correlations
,”
Ann. Inst. Henri Poincare, Probab. Stat.
58
(
1
),
220
247
(
2022
); arXiv:2007.08272 [math.PR].
33.
G.
Carinci
,
C.
Franceschini
, and
W.
Groenevelt
, “
q—Orthogonal dualities for asymmetric particle systems
,”
Electron. J. Probab.
26
,
1
38
(
2021
); arXiv:2003.07837 [math.PR].
34.
C.
Franceschini
,
J.
Kuan
, and
Z.
Zhou
, “
Orthogonal polynomial duality and unitary symmetries of multi–species ASEP(q, θ) and higher–spin vertex models via ∗–bialgebra structure of higher rank quantum groups
,” arXiv:2209.03531 [math.PR] (
2022
).
35.
T.
Sasamoto
and
M.
Wadati
, “
One-dimensional asymmetric diffusion model without exclusion
,”
Phys. Rev. E
58
,
4181
4190
(
1998
).
36.
A. M.
Povolotsky
, “
On the integrability of zero-range chipping models with factorized steady states
,”
J. Phys. A: Math. Theor.
46
(
46
),
465205
(
2013
); arXiv:1308.3250 [math-ph].
37.
G.
Barraquand
and
I.
Corwin
, “
The q-Hahn asymmetric exclusion process
,”
Ann. Appl. Probab.
26
(
4
),
2304
2356
(
2016
); arXiv:1501.03445 [math.PR].
38.
R.
Frassek
, “
Integrable boundaries for the q-Hahn process
,”
J. Phys. A: Math. Theor.
55
(
40
),
404008
(
2022
); arXiv:2205.10512 [math-ph].
39.
R.
Frassek
, “
Eigenstates of triangularisable open XXX spin chains and closed-form solutions for the steadystate of the open SSEP
,”
J. Stat. Mech.
2005
,
053104
(
2020
).
40.
R.
Frassek
,
C.
Giardinà
, and
J.
Kurchan
, “
Duality and hidden equilibrium in transport models
,”
SciPostPhys.
9
,
054
(
2020
).
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