1549380323-Statistical Mechanics Theory and Molecular Simulation

44 Classical mechanics ly=−ωx ∑n i=1 miyixi+ωy ∑n i=1 mi(r^2 i−y^2 i)−ωz ∑n i=1 miyizi lz=−ωx ∑n i=1 mizixi−ωy ∑n i=1 miziyi+ωz ...

Rigid body motion 45 I 2 ω ̇ 2 −ω 3 ω 1 (I 3 −I 1 ) =τ 2 I 3 ω ̇ 3 −ω 1 ω 2 (I 1 −I 2 ) =τ 3. (1.11.35) These are known as theri ...

46 Classical mechanics For a rigid body in three dimensions, we require four variables, (q 1 ,q 2 ,q 3 ,q 4 ), the quaternions, ...

Non-Hamiltonian systems 47 ω ̇y= τy Iyy + (Izz−Ixx) Iyy ωzωx ω ̇z= τz Izz + (Ixx−Iyy) Izz ωxωy. (1.11.44) Here,ω= (0,ωx,ωy,ωz) a ...

48 Classical mechanics the entire system and determine the evolution of the physical subsystem. However, we are rarely intereste ...

Problems 49 p ̇=−mω^2 x−ζ p m . (1.12.4) It can be seen that this dynamical system has a non-vanishing compressibility κ(x,p) = ...

50 Classical mechanics 1.13 Problems 1.1. Solve the equations of motion given arbitrary initial conditions for a one- dimensiona ...

Problems 51 1.6. A water molecule H 2 O is subject to an external potential. Let the positions of the three atoms be denotedrO,r ...

52 Classical mechanics (^1) q 2 q l 2 l 1 m 1 m 2 Fig. 1.13Schematic of a diatomic molecule bound to a flat surface. 1.10. Consi ...

Problems 53 interacting with a static electromagnetic field is given by H= ∑N i=1 (pi−qiA(ri)/c)^2 2 mi + ∑N i=1 qiφ(ri) + N∑− 1 ...

2 Theoretical foundations of classical statistical mechanics 2.1 Overview The field of thermodynamics began in precursory form w ...

Laws of thermodynamics 55 The highly nonlinear character of the forces in realistic systems means that an analyt- ical solution ...

56 Theoretical foundations 2.2 The laws of thermodynamics Our discussion of the laws of thermodynamics will make no referenceto ...

Laws of thermodynamics 57 P V T 2 1 Fig. 2.1The thermodynamic state space defined by the variablesP,V, andTwith two paths (solid ...

58 Theoretical foundations first law states thatin any thermodynamic transformation, if a system absorbs an amount of heat∆Qand ...

Laws of thermodynamics 59 2.2.2 The second law of thermodynamics Before discussing the second law of thermodynamics, it is usefu ...

60 Theoretical foundations −Wnet= ∆Q=Qh+Ql. (2.2.11) The efficiency of any engineǫis defined as the ratio of the net work output ...

Laws of thermodynamics 61 Eqn. (2.2.18) indicates that there is a quantity ∆Qrev/T whose change over the closed cycle is 0. The ...

62 Theoretical foundations 2.2.3 The third law of thermodynamics As with any state function, it is only possible to measure chan ...

Ensembles 63 The idea that the macroscopic observables of a system are not sensitive to pre- cise microscopic details is the bas ...