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By Michael Reed, Barry Simon

BESTSELLER of the XXth Century in Mathematical Physics voted on by way of members of the XIIIth overseas Congress on Mathematical Physics
This revision will make this e-book extra appealing as a textbook in sensible research. extra refinement of insurance of actual subject matters also will toughen its well-established use as a direction e-book in mathematical physics.

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Starting with Eq. (92), x(t) = x(0) + v(0) 1 (1 − e−βt ) + β β t 1 − eβ(t −t) F(t )dt (118) 0 and multiplying by the force at time t , we have x(t)F(t ) = x(0)F(t ) + v(0) 1 (1 − e−βt )F(t ) + β β t 0 1 − eβ(t −t) F(t )F(t )dt (119) 24 L. Cohen Taking expectation values x(t)F(t ) = 2D β t 1 − eβ(t −t) δ(t − t )dt (120) 0 which evaluates to x(t)F(t ) = 2D β 1 − e−β(t−t ) 0 < t < t 0 otherwise (121) Note that the correlation between position and the force at a later time is zero, x(t)F(t + τ ) = 0 τ >0 (122) The reason for this is that if the force acts at a time later than the position time, it will have had no affect on the position.

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