Playing with Non Equilibrium, the Maximum Entropy Way Mandar M. Inamdar, Effrosyni Seitaridou & Rob Phillips California Institute of Technology & Kingshuk Ghosh & Ken Dill University of California, San Francisco The theory is out there. Our goal is to apply the theory to very simple models in biology, physics or chemistry to gain insights into the dynamics of the problems in these areas.

Goal is to find the probabilities of these trajectories. Dynamics and Time Trajectories We want to address the experiments involving time trajectories and time varying driving forces. RNA unfolding by mechanical force Liphardt et al., 2001 People find many different realisation of trajectories. We are interested in finding the probabilities of trajectories. Example to show how things work. Motor with steps. Can get rid of upper left. Goal is to find the probabilities of these trajectories. Potassium Conductance Llano et al., 1988

Principle of Maximum Entropy Partial Information {A1,…,Ak} The system can be a physical system, image, spectroscopic data, or even a language sample. We have insufficient information {A1,…Ak} about the system. Information Entropy with constraints Maximize entropy to get px Maximum entropy is way of making predictions with minimum information. Use license plate example in this. Principle of maximum entropy gives us the least biased probability distribution of the various states x of the system. © J. Skilling

Why use maximum entropy? Image Processing for Astronomy Processing Raman Spectra © J. Skilling Maxent is extremely versatile, with multifarious applications. Used to give best estimates with incomplete information. Do not go into too much details. There is just one idea on this page. Do not bang on the head of may people. Language processing Berger et al., 1996

Application of Maxent to dynamics 1. Define the model, trajectories and microscopic dynamics. Goal is to find the probability of trajectory p Г . We write down the trajectory entropy 2. Identify Constraints {A1,…,Ak} 3. Write the entropy over trajectories Г Just use one set of trajectories. Just use the kinematics of the problem. Use maxent on the set of trajectories. Get rid of equations. 4. Maximize entropy to obtain pГ

Ehrenfest’s Dog-Flea Model Dogs I and II are separated by Δx. Flea hops from I to II, and from II to I with probability p1 and p2 in time Δt. The trajectory is defined by the set {m1, m2}. We can write entropy over trajectories. If p1 = p2 = p, on maximizing the entropy, we will recover Fick’s law for flux: The total number of trajectories at any instant is Fick’s law emerges from summing over all possible jumps. Do not show too many equations.

Bad Actors The macroscopic flux has a certain direction. The trajectories which go against the average flux are termed as bad-actors. The fraction of bad actors is maximum at equilibrium.

Potency The 45 degree line is where m1= m2, i.e., flux is zero. The shaded strip of thickness h is the zone in which the flux is around zero. The non-shaded area is where the flux brings about macroscopic change. Remove the equation. Explain the physical idea. The notion of potency denotes how many trajectories change the macroscopic flux. The total number of potent trajectories is least at equilibrium.

Constraints: Power consumption, and dissipation Other Applications Constraints: Power consumption, and dissipation Do the motor. Can ditch this thing. Processive motion of Myosin V K+ ion channel Rief et al., 2000

Conclusions Acknowledgements Application of maximum entropy principle to Dog-Flea model gives Fick’s law, while providing us with fluctuations. Introduction of the notion of potency and bad-actors , which is novel in the non-equilibrium setting. A different perspective on one-step and two-step processes. Acknowledgements Make a collage. Here.Ditch the collage Rob Phillips & Effrosyni Seitaridou, Caltech Kingshuk Ghosh & Ken Dill, UCSF