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**Monte Carlo Methods and Statistical Physics**

Mathematical Biology Lecture 4 James A. Glazier (Partially Based on Koonin and Meredith, Computational Physics, Chapter 8)

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**Two Basic Applications:**

Monte Carlo Methods Use Statistical Physics Techniques to Solve Problems that are Difficult or Inconvenient to Solve Deterministically. Two Basic Applications: Evaluation of Complex Multidimensional Integrals (e.g. in Statistical Mechanics) [1950s] Optimization of Problems where the Deterministic Problem is Algorithmically Hard (NP Complete—e.g. the Traveling Salesman Problem) [1970s]. Both Applications Important in Biology.

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**Example: Thermodynamic Partition Function**

For a Gas of N Atoms at Temperature 1/b, Interacting Pairwise through a Potential V(r), the Partition Function: Suppose We need to Evaluate Z Numerically with 10 Steps/Integration. Then Have 103N Exponentials to Evaluate. The Current Fastest Computer is About 1012 Operations/Second. One Year ~ 3 x 107 Seconds. So One Year ~ 3 x 1019 Operations. In One Year Could Evaluate Z for about 7 atoms! This Result is Pretty Hopeless. There Must Be a Better Way.

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**Normal Deterministic Integration**

Consider the Integral: Subdivide [0,1] into N Evenly Spaced Intervals of width Dx=1/N. Then:

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**Error Estimate—Continued**

2) Convergence is Slow: while for Normal Deterministic Integration: However, Comparison Isn’t Fair. Suppose You Fix the Number of Subdomains in the Integral to be N. In d Dimensions Each Deterministic Sub-Integral has N1/d Intervals. So the Net Error is So, if d>4 the Monte Carlo Method is Better!

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**Error Estimate How Good is the Estimate?**

For a constant Function, the Error is 0 for Both Deterministic and Monte Carlo Integration. Two Rather Strange Consequences: In Normal Integration, Error is 0 for Straight Lines. In Monte Carlo Integration, Errors Differ for Straight Lines Depending on Slope (Worse for Steeper Lines). If

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**Monte Carlo Integration**

Use the Idea of the Integral as an Average: Before We Solved by Subdividing [0,1] into Evenly Spaced Intervals, but could Equally Well Pick Positions Where We Evaluate f(x) Randomly: Chosen to be Uniform Random. So: Approximates I Note: Need a Good Random Number Generator for this Method to Work. See (Vetterling, Press…, Numerical Recipies)

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Pathology Like Normal Integration, Monte Carlo Integration Can Have Problems. Suppose You have N Delta Functions Scattered over the Unit Interval. However, the Probability of Hitting a Delta Function is 0, so IN=0. For Sharply-Peaked Functions, the Random Sample is a Bad Estimate (Standard Numerical Integration doesn’t Work Well Either)

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Weight Functions Can Improve Estimates by Picking the ‘Random’ Points Intelligently, to Have More Points Where f(x) is Large and Few Where f(x) is Small. Let w(x) be a Weight Function Such That: For Deterministic Integration, the Weight Function has No Effect: Let: Then: Alternatively, Pick: So All We have to do is Pick x According to the Distribution w(x) and Divide f(x) by that Distribution:

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**Weight Functions—Continued**

If Why Not Just Let w(x)= f(x)? Then Need to Solve the Integral to Invert y(x) to Obtain x(y) or to Pick x According to w(x). But Stripping Linear Drift is Easy and Always Helps. In d dimensions have: So: Which is Hard to Invert, so Need to Pick Directly (Though, Again Can Strip Drift).

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Example Let And Then When You Can’t Invert y(x) Refer to Large Literature on How to Generate With the Needed Distribution.

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Metropolis Algorithm Originally a Way to Derive Statistics for Canonical Ensemble in Statistical Mechanics. A Way to Pick the According to the Weight Function in a very high dimensional space. Idea: Pick any x0 and do a Random Walk: Subject to Constraints Such that the Probability of a Walker at has: Problems: 1) Convergence can be Very Slow. 2) Result can be Wrong. 3) Variance Not Known.

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Algorithm For any State Generate a New Trial State Usually (Not Necessary) Assume that is Not Too Far From I.e. that it lies within a ball of Radius d >0 of : Let: If r≥1 then Accept the Trial: If r<1 then Accept the Trial with Probability r. I.e. Pick a Random Number [0,1]. If <r then Accept: Otherwise Reject: Repeat.

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**Problems May Not Sample Entire Space.**

If d Too Small Explore only Small Region Around . If d Too Big Probability of Acceptance Near 0. Inefficient. If Regions of High w Linked by Regions of Very Low w Never See Other Regions. If w Sharply Peaked Tend to Get Stuck Near Maximum. Sequence of Not Statistically Independent, so Cannot Estimate Error. Fixes: Use Multiple Replicas. Many Different , Which Together Sample the Whole Space. Pick d So that the Acceptance Probability is ~ ½ (Optimal). Run Many Steps Before Starting to Sample.

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**Convergence Theorem Theorem:**

Proof: Consider Many Independent Walkers Starting from Every Possible Point and Let Them Run For a Long Time. Let be the Density of Walkers at Point . Consider Two Points and Let be the Probability for a Single Walker to Jump from to . Rate of Jumps from to is: So the Net Change in is:

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**Convergence Theorem—Continued**

At Equilibrium: So if And if So Always Tends to its Equilibrium Value Monotonically and the Rate of Convergence is Linear in the Deviation: Implies that System is Perfectly Damped. This Result is the Fundamental Justification for Using the Metropolis Algorithm to Calculate Nonequilibrium and Kinetic Phenomena.

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**Convergence Theorem—Conclusion**

Need to Evaluate If is Allowed, So is (Detailed Balance). I.e. So While if So Always. Normalizing by the Total Number of Walkers □ Note that This Result is Independent of How You Choose Given , As Long as Your Algorithm Has Nonzero Transition Probabilities for All Initial Conditions and Obeys Detailed Balance (Second Line Above).

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