复旦大学：《热力学与物理统计 Thermodynamics & Statistical Physics》PPT课件（金晓峰）

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NGS IN PHYSICS: KINETIC THEORY MAXWELL: THE DYNAMICAL THEORY OF GASES so that the probability is independent of that is, all directions of rebound are equally likely Let n be the whole number of particles. Let x, y, z be the com- ponents of the velocity of each particle in three rectangular direc Prop. Ill. Given the direction and magnitude of the velocities of tions, and let the number of particles for which x lies between x and two spheres before impact, and the line of centres at impact; to find x +dx, be Nfr)dx, where f() is a function of x to be determined. the velocities after impact The number of particles for which y lies between y and y dy Let OA, OB represent the velocities before impact, so that if there will be NfG)dy; and the number for which z lies between z and had been no action between the bodies they would have been at a z dz will be Nf(z)dz, where f always stands for the same functio and b at the end of a second, Join AB, and let g be their centre of Now the existence of the velocity x does not in any way affect that of the velocities y or z, since these are all at right angles to each other and independent, so that the number of particles whose velocity lies between x and x dx, and also between y and y +dy, and also between z and z +dz,is Nf(r)fv)f(z)dx dy dz If we suppose the N particles to start c心 at the same instant, then this will be the number of volume dr dy dz)after unit of time, and the gravity, the position of which is not affected by their mutual action to unit of volume will be Draw GN parallel to the line of centres at impact(not necessarily in the plane AOB). Draw aGb in the plane AGN, making NGa NGA N(x)U)/(z) and Ga= GA and Gb=GB; then by Prop. I. Ga and Gb will be But the directions of the coordinates are perfectly arbitrary, and the velocities relative to G; and compounding these with OG, therefore this number must depend on the distance from the origin have Oa and Ob for the true velocities after impact alone that is By Prop. Il. all directions of the line aGb are equally probable. fx)(υ)(2)=x2+y2+z2) It appears therefore that the velocity after impact is compounded of locity of the sphere relative ravity, and of a velocity equal to the he velocity of the centre of Solving this functional equation, we to the centre of gravity, which may with f(r=Ce, P(r2)=C3etr qual probability be in any direction whatever If a great many equal spherical particles were in motion in a If we make A positive, the number of particles will increase with perfectly elastic vessel, collisions would take place among the the velocity, and we should find the whole number of particles particles, and their velocities would be altered at every collision; so infinite. We therefore make A negative and equal to 1/a, so that that after a certain time the uis vipa will be divided among the the number between x and x +dx is particles according to some regular law, the average number of certain limits bei able, though the velocity of each particle changes at every collision Integrating from x = -oo to x =+oo, we find the whole number Prop. IV. To find the average number of pat of particles lie between given limits, after a great number of collisions among a great number of equal particles Ncvπ=N,∴C=

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