1549380323-Statistical Mechanics Theory and Molecular Simulation

4 Classical mechanics ̈r= 0. (1.2.8) The straight line motion of eqn. (1.2.7) is, in fact, the unique solution of eqn. (1.2.8) f ...

Phase space 5 itself to an analytical solution could be introduced. Although often of limited util- ity, important physical insi ...

6 Classical mechanics instanttin time, all of the information about the system is specified by 6Nnumbers (or 2dNinddimensions). ...

Phase space 7 Another important example of a phase space trajectory is that ofa simple harmonic oscillator, for which the force ...

8 Classical mechanics conditions give rise to different values ofC, which changes the size of the ellipse. Changing the mass and ...

Lagrangian formulation 9 one particular initial rolling speed in which the ball can just climb to the top of the hill and come t ...

10 Classical mechanics 1.4 Lagrangian formulation of classical mechanics: A general framework for Newton’s laws Statistical mech ...

Lagrangian formulation 11 It can be easily verified that substitution of eqn. (1.4.5) into eqn. (1.4.6) gives eqn. (1.2.10): ∂L ...

12 Classical mechanics In order to verify thatEis a constant, we need only show that dE/dt= 0. Differen- tiating eqn. (1.4.11) w ...

Lagrangian formulation 13 = 1 2 ∑^3 N α=1 ∑^3 N β=1 Gαβ(q 1 ,...,q 3 N) ̇qαq ̇β, (1.4.16) where Gαβ(q 1 ,...,q 3 N) = ∑N i=1 mi ...

14 Classical mechanics 1.4.1 Example: Motion in a central potential Consider a single particle in three dimensions subject to a ...

Lagrangian formulation 15 choosing a coordinate frame in which thezaxis lies along the direction ofl. In such a frame, the motio ...

16 Classical mechanics m 2 ̈r 2 =U′(|r 1 −r 2 |) r 1 −r 2 |r 1 −r 2 | , (1.4.34) a more natural set of coordinates can be chosen ...

Legendre transforms 17 x x 0 slope = f ’(x 0 ) f(x) + c f(x) Fig. 1.6Depiction of the Legendre transform. clear, is thats 0 , be ...

18 Classical mechanics The generalization of the Legendre transform to a functionfofnvariablesx 1 ,...,xn is straightforward. In ...

Hamiltonian formulation 19 =− ∑N i=1 p^2 i 2 mi −U(r 1 ,...,rN). (1.6.2) The function−L ̃(r 1 ,...,rN,p 1 ,...,pN) is known as t ...

20 Classical mechanics +U(r 1 (q 1 ,...,q 3 N),...,rN(q 1 ,...,q 3 N)). (1.6.10) Given the Hamiltonian (as a Legendre transform ...

Hamiltonian formulation 21 = ∑^3 N α=1 [ ∂H ∂qα ∂H ∂pα − ∂H ∂pα ∂H ∂qα ] = 0, (1.6.15) where the second line follows from Hamilt ...

22 Classical mechanics = ∑^3 N α=1 [ ∂a ∂qα q ̇α+ ∂a ∂pα p ̇α ] = ∑^3 N α=1 [ ∂a ∂qα ∂H ∂pα − ∂a ∂pα ∂H ∂qα ] ≡ {a,H}. (1.6.19) ...

Hamiltonian formulation 23 Another fundamental property of Hamilton’s equations is known asthe condi- tion ofphase space incompr ...