﻿ 两种坐标系下速度(率)、能量、动量的分布 Velocity, Energy and Momentum’s Distribution Based on Two Coordinate System

Modern Physics
Vol.05 No.01(2015), Article ID:14763,5 pages
10.12677/MP.2015.51003

Velocity, Energy and Momentum’s Distribution Based on Two Coordinate System

Jincheng He, Xiaobo Li, Feng Luan

College of Engineering Technology, Chengdu University of Technology, Leshan

Email: 759155750@qq.com

Received: Jan. 10th, 2015; accepted: Jan. 22nd, 2015; published: Jan. 28th, 2015

ABSTRACT

In this paper, the distribution of velocity in Cartesian coordinates and spherical coordinates deduces on the basis of the distribution of energy and momentum, and calculates the most probable speed in both coordinates and the most probable energy, average speed, average energy and average momentum and other related physical quantities. Through the analysis results, we found that: in both the coordinates, the calculation result was consistent with scalar, vector calculation results were different. This paper also studied the scope of Maxwell distribution and the reason why the most probable energy Maxwell energy distribution corresponding speed and the most probable speed of Maxwell velocity distribution values were not equal.

Keywords:Maxwell Distribution, Both Coordinates, Velocity, Energy, Momentum

Email: 759155750@qq.com

1. 引言

2. 推导麦克斯韦速度分布函数

2.1. 推导直角坐标系下麦克斯韦分布

(1)

Figure 1. Cartesian coordinates

(1)式左右两端对y,z分别在积分求得在x方向的麦克斯韦速度分布如下：

(2)

(3)

(4)

(5)

(6)

(7)

(8)

i方向的最概然速度，令：

(9)

i方向的平均速度

(10)

i方向的平均值

(11)

(12)

i方向最概然动量，令：

(13)

i方向动量的平均值

(14)

i方向的平均值

(15)

2.2. 推导球坐标系下麦克斯韦分布

(16)

(17)

(18)

(19)

(20)

Figure 2. Spherical coordinates

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

2.3. 两种坐标系下计算结果比较

3. 结论

1) 平均能量、方均根速度(速率)与方均根动量都是标量。标量在直角坐标和球坐标下得到的结果是一致的；

2) 最概然速度、平均速度、最概然动量与平均动量都是直角坐标下的矢量；而最概然速率、平均速率、最概然动量与平均动量都是球坐标下的标量。在直角坐标的矢量和球坐标下标量得到的结果是不一致的。

3) 麦克斯韦分布适用范围有一个发展过程。它不仅适用于理想气体的速度或速率分布，同时也适用于非理想气体和多元气体；后来又被推广到固体、液体或任何经典粒子系统；再后来讨论了相对论效应

Table 1. The calculation of each physical in both coordinates

4) 最概然速率对应的能量与最概然能量不相等，这一结果可能引起读者疑惑现做如下说明：才表示速率为，对应在速率附近厚度球壳内的粒子数为

Velocity, Energy and Momentum’s Distribution Based on Two Coordinate System. 现代物理,01,23-28. doi: 10.12677/MP.2015.51003

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