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# lecture17 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 17 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 17 In last lecture, we discussed the N-particle distribution function. We have obtained ( ) ( ) ) n ( ) n ( ) n ( ) n ( N , , t f Systems . No p r p r = in a small volume of the phase space, r (n) p (n) , where r (n) = r 1 r 2 r N = r (1) r (2) r (n) and p (n) = p 1 p 2 p N = p (1) p (2) p (n) . The time evolution of f (N) (t, r (n) , p (n) ) in the phase space is governed by the Liouville equation , 0 p f p r f r t f n 1 i ) i ( ) N ( ) i ( n 1 i ) i ( ) N ( ) i ( ) N ( = × + × + = = ± ± which can be derived based on the fact that the traces of systems in the ensemble do not intersect. Note: The number of the degree of freedom in the phase space is normally very big. For 3D cases, it is 6N A =6 × 6.02E23. The Liouville equation involves a huge number of variables, which makes its impractical in terms of the boundary and initial conditions, as well as analytical and numerical solutions. One way to simplify the Liouville equation is to consider one particle in a system. This is a representative particle having coordinate r 1 and momentum p 1 , each of the vectors has m components, i.e., m-degrees of freedom. We introduce a one-particle distribution function by averaging the N-particle distribution function over the rest (N- 1) particles in the system, r (i) p (i) ensemble at t=0 Flow-line control volume r (i) r (i) + r (i)

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2.57 Fall 2004 – Lecture 17 2 ( ) ( ) N N p p r r p r p r d d d d , , t f )! 1 N ( ! N , , t f 2 2 ) n ( ) n ( ) N ( 1 1 ) 1 ( = . For simplicity in notation, we will drop the subscript 1 and understand ( r , p ) as the coordinates and momenta of one particle. Since f (N) (t, r (n) , p (n) ) represents the number density of systems having generalized coordinates ( r (n) , p (n) ) in the ensemble, the one particle distribution function represents number density of systems having ( r , p ), f(t, r , p )d 3 r d 3 p =number of systems in d 3 r d 3 p . With the introduction of the averaging method to obtain the one-particle distribution function, one can start from the Liouville equation and carry out the averaging over the space and momentum coordinates of the other (N-1) particles. This procedure leads to c t f f dt d f dt d t f = + + p r p r where the subscripts ( r and p ) in the gradient operators represent the variables of the gradient. The scattering term c f t will be discussed in details later in this lecture. The above equation is the Boltzmann equation or Boltzmann transport equation. Note: The derivative d dt r has the meaning of velocity, while d dt p denotes force. Consider the collision process between two particles as shown in the above figure. After the collision, the energy and the velocity of each particle may change. Clearly, the collision is a time-dependent process. The rigorous way of dealing the collision process is to solve the corresponding time-dependent Schrödinger equation for the combined system made of both particles. This is, however, usually very complicated and not practical.
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lecture17 - 2.57 Nano-to-Macro Transport Processes Fall...

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