@article{10023,
  abstract     = {We study the temporal dissipation of variance and relative entropy for ergodic Markov Chains in continuous time, and compute explicitly the corresponding dissipation rates. These are identified, as is well known, in the case of the variance in terms of an appropriate Hilbertian norm; and in the case of the relative entropy, in terms of a Dirichlet form which morphs into a version of the familiar Fisher information under conditions of detailed balance. Here we obtain trajectorial versions of these results, valid along almost every path of the random motion and most transparent in the backwards direction of time. Martingale arguments and time reversal play crucial roles, as in the recent work of Karatzas, Schachermayer and Tschiderer for conservative diffusions. Extensions are developed to general “convex divergences” and to countable state-spaces. The steepest descent and gradient flow properties for the variance, the relative entropy, and appropriate generalizations, are studied along with their respective geometries under conditions of detailed balance, leading to a very direct proof for the HWI inequality of Otto and Villani in the present context.},
  author       = {Karatzas, Ioannis and Maas, Jan and Schachermayer, Walter},
  issn         = {1526-7555},
  journal      = {Communications in Information and Systems},
  keywords     = {Markov Chain, relative entropy, time reversal, steepest descent, gradient flow},
  number       = {4},
  pages        = {481--536},
  publisher    = {International Press},
  title        = {{Trajectorial dissipation and gradient flow for the relative entropy in Markov chains}},
  doi          = {10.4310/CIS.2021.v21.n4.a1},
  volume       = {21},
  year         = {2021},
}

@misc{5587,
  abstract     = {Supporting material to the article 
STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH

boundscoli.dat
Flux Bounds of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium. 

polcoli.dat
Matrix enconding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium, 
obtained from the soichiometric matrix by standard linear algebra  (reduced row echelon form).

ellis.dat
Approximate Lowner-John ellipsoid rounding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium
obtained with the Lovasz method.

point0.dat
Center of the approximate Lowner-John ellipsoid rounding the polytope of the E. coli catabolic core model iAF1260 in a glucose limited minimal medium
obtained with the Lovasz method.

lovasz.cpp  
This c++ code file receives in input the polytope of the feasible steady states of a metabolic network, 
(matrix and bounds), and it gives in output an approximate Lowner-John ellipsoid rounding the polytope
with the Lovasz method 
NB inputs are referred by defaults to the catabolic core of the E.Coli network iAF1260. 
For further details we refer to  PLoS ONE 10.4 e0122670 (2015).

sampleHRnew.cpp  
This c++ code file receives in input the polytope of the feasible steady states of a metabolic network, 
(matrix and bounds), the ellipsoid rounding the polytope, a point inside and  
it gives in output a max entropy sampling at fixed average growth rate 
of the steady states by performing an Hit-and-Run Monte Carlo Markov chain.
NB inputs are referred by defaults to the catabolic core of the E.Coli network iAF1260. 
For further details we refer to  PLoS ONE 10.4 e0122670 (2015).},
  author       = {De Martino, Daniele and Tkacik, Gasper},
  keywords     = {metabolic networks, e.coli core, maximum entropy, monte carlo markov chain sampling, ellipsoidal rounding},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Supporting materials "STATISTICAL MECHANICS FOR METABOLIC NETWORKS IN STEADY-STATE GROWTH"}},
  doi          = {10.15479/AT:ISTA:62},
  year         = {2018},
}