Gillespie_CMP_handouts

Gillespie_CMP_handouts - 1 Making Differential...

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Unformatted text preview: 1 Making Differential Equations &on-Deterministic Dan Gillespie Dan T Gillespie Consulting [email protected] Current Support : Caltech (NIGMS, NIH) University of California at Santa Barbara (NIH) Past Support : Caltech (DARPA/AFOSR, Beckman/BNCM)) University of California at Santa Barbara (DOE) Molecular Sciences Institute (Sandia/DOE) Office of &aval Research It’s everywhere … ( ) ( ) ( ), dX t A X t t dt = . (1) Newton’s 2 nd Law: 2 2 d x F m dt = ⇔ ( ) ( ) ( ) ( ), ( ), ( ) dx t v t dt F x t v t t dv t dt m = = Defining 1 ( ) ( ) ( ) and ( ( ), ) ( ) ( ( ), ( ), ) v t x t X t A X t t v t m F x t v t t − ≡ ≡ , we get (1). • If the independent variable t represents time , we call X a process . - Time is a very special variable.- It always increases . 2 the Past the Present the Future ( ) ( ) ( ), dX t A X t t dt = (1) Another way of writing (1), which shows explicitly that it determines the future values of X in terms of the present value of X , is this: ( ) ( ) ( ( ), ) ( ) ( 0) X t t X t A X t t t o t t Δ Δ Δ Δ + = + + ≥ , where ( ) o t Δ satisfies lim ( ) t o t t Δ Δ Δ + → = . Or more simply . . . ( ) ( ) ( ( ), ) X t dt X t A X t t dt + = + . (2) In (2), dt is an ordinary real variable , distinct from t, which is confined to [0, ε ), where ε is an arbitrarily small positive number. h In other words . . . dt is “a little bit of t .” time ( ) ( ) ( ), dX t A X t t dt = (1) ( ) ( ) ( ( ), ) X t dt X t A X t t dt + = + . (2) • In any real situation, (2) will be a “sufficiently accurate” expression of (1) for all [0, ] dt ε ∈ for some sufficiently small ε > . • (1) and (2) are logically equivalent , because they define the same process X . • (2) has the form of an update formula for X : It’s a recipe for computing, from the value of X at the present time t , its value at the later time t dt + for all [0, ] dt ε ∈ . 3 ( ) ( ) ( ( ), ) X t dt X t A X t t dt + = + (2) The process X defined by (2) has three obvious properties : • X is continuous : ( ) ( ) X t dt X t + → as dt → . • X is memoryless (Markov) : To compute any future value ( ) X t dt + , (2) needs only ( ) X t ; it does not need any past values ( ) X t t ′ < . • X is deterministic : The update formula fixes the future value ( ) X t dt + unequivocally. A Question : Is (2) the only form for an update formula for a process that is continuous and memoryless and deterministic ? How about: ( ) ( ) ( ( ), ) X t dt X t A X t t dt + = + . (*) • There is no problem taking the square root of any number in [0, ) ε . • X as defined by (*) is not differentiable . But so what?...
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Gillespie_CMP_handouts - 1 Making Differential...

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