^{1,a)}and Angelo Vulpiani

^{2}

### Abstract

We propose a general method for simplifying master equations by eliminating from the description rapidly evolving states. The physical recipe we impose is the suppression of these states and a renormalization of the rates of all the surviving states. In some cases, this decimation procedure can be analytically carried out and is consistent with other analytical approaches, such as in the problem of the random walk in a double well potential. We discuss the application of our method to nontrivial examples: diffusion in a lattice with defects and a model of an enzymatic reaction outside the steady state regime.

We are grateful to M. Cencini and A. Puglisi for useful remarks and a detailed reading of the manuscript. A.V. wishes to thank Universitad de las Islas Baleares (Palma de Mallorca, Spain) for hospitality during the first stage of this work. S.P. wishes to thank A. D. Jackson for stimulating discussions and help with the commutativity argument.

I. INTRODUCTION

II. THE METHOD

A. Decimation procedure

B. Commutativity and adiabatic approximations

III. ONE DIMENSIONAL EXAMPLES

A. Double well potential

B. Random walk with defects

IV. ENZYMATIC REACTIONS

V. CONCLUSIONS

### Key Topics

- Eigenvalues
- 11.0
- Enzymes
- 9.0
- Random walks
- 8.0
- Enzyme kinetics
- 7.0
- Chemical kinetics
- 6.0

## Figures

Schematic example: A particle confined in a four-well potential.

Schematic example: A particle confined in a four-well potential.

Schematic example of the transition rates before and after the decimation in a master equation with four states. Here, the state on the left is a fast state and is eliminated from the description. On the right, we represent the renormalized transition in terms of the original rates. The new transition rate from to contains the original one plus an additional contribution coming from the elimination of . The transition rate from to , equal to 0 in the original graph, contains only the effect of the decimation.

Schematic example of the transition rates before and after the decimation in a master equation with four states. Here, the state on the left is a fast state and is eliminated from the description. On the right, we represent the renormalized transition in terms of the original rates. The new transition rate from to contains the original one plus an additional contribution coming from the elimination of . The transition rate from to , equal to 0 in the original graph, contains only the effect of the decimation.

Particle in a discrete double well potential. and are the states corresponding to the two minima of the potential.

Particle in a discrete double well potential. and are the states corresponding to the two minima of the potential.

Eigenvalues listed in decreasing order of a random walk with defect. The number of state is . Circles: No defects, the jump rate is . Squares: With probability , the sites have defects and their jump probability is . Diamonds: Random walk with defects after applying the decimation scheme to all the fast states. All the eigenvalues are real. The difference between the figure and the inset is just the axis scale. Notice in the main figure that the first 100 eigenvalues of the decimated problem follow very closely the case with the defects. In the inset, the eigenvalues corresponding to the fast states are evident; notice that their number is the same as the average number of defects .

Eigenvalues listed in decreasing order of a random walk with defect. The number of state is . Circles: No defects, the jump rate is . Squares: With probability , the sites have defects and their jump probability is . Diamonds: Random walk with defects after applying the decimation scheme to all the fast states. All the eigenvalues are real. The difference between the figure and the inset is just the axis scale. Notice in the main figure that the first 100 eigenvalues of the decimated problem follow very closely the case with the defects. In the inset, the eigenvalues corresponding to the fast states are evident; notice that their number is the same as the average number of defects .

Correlation functions. The system is the same as Fig. 4 prepared in the initial state , and the probability of being in state , averaged over realization, is plotted as a function of time. The three curves are of (continuous) the system without defects, (dashed) system with defects, and (dot dashed) the decimated system.

Correlation functions. The system is the same as Fig. 4 prepared in the initial state , and the probability of being in state , averaged over realization, is plotted as a function of time. The three curves are of (continuous) the system without defects, (dashed) system with defects, and (dot dashed) the decimated system.

Simulations of the master equation corresponding to the enzymatic reaction of Eq. (21). In all simulations, we start with of free substrate and let the system evolve: the rate as a function of the number of molecules is evaluated by averaging over realizations of the process. The numbers of enzymes are in the top figures and in the bottom figures. The reaction rates are (left) , and (right) , . The lines in the right figures are barely visible since the points fall very close to them.

Simulations of the master equation corresponding to the enzymatic reaction of Eq. (21). In all simulations, we start with of free substrate and let the system evolve: the rate as a function of the number of molecules is evaluated by averaging over realizations of the process. The numbers of enzymes are in the top figures and in the bottom figures. The reaction rates are (left) , and (right) , . The lines in the right figures are barely visible since the points fall very close to them.

Product variance as a function of time, each averaged over realizations. The four figures correspond to the same parameter choices of Fig. 6. Continuous lines correspond to simulation of the full master equation, dashed lines are simulations of the reduced processes defined by Eqs. (25)–(27).

Product variance as a function of time, each averaged over realizations. The four figures correspond to the same parameter choices of Fig. 6. Continuous lines correspond to simulation of the full master equation, dashed lines are simulations of the reduced processes defined by Eqs. (25)–(27).

Two states ( and , black) separates by a cluster of N states (white) to be decimated.

Two states ( and , black) separates by a cluster of N states (white) to be decimated.

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