[PDF] [PDF] MAS864: Derivation of 2D Boltzmann Distribution - MIT Fab Lab

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MAS.864: Derivation of 2D Boltzmann Distribution

Dhaval Adjodah

MIT

May 16, 2011

From the Kinetic Theory of gases, the general form of the probability density function of the velocity component of a gas particle is of the form p(vi) =AeBv2i:(1) Since we are in 2 dimensions, the speed of a particle is v=qv

2x+v2y:(2)

with dierential elementvdvd. Integratingfrom 0 to 2, we can see that the speed probability density function is p(v) = 2vA2eBv2i:(3)

Using normalization, and the standard integralZ1

0 xeBx2=12B;(4)

One can show that

A=rB :(5) Hence, the only free parameter to be determined isB, which will now be pursued Given that the average kinetic energy of a particle with 2 degrees of freedom (in the

2-dimensional case for a spherical particle) iskTwherekis the Boltzmann constant and

Tthe thermodynamic temperature, it follows that1

2 mv2=Z 1 012 mv2p(v) =kT:(6) Integrating above equation and solving forB, one can determine thatA=pm

2kTand

B=m2kT. This leads to the probability density function to be p(v) =mkT vem2kTv2(7) v

2ev222(8)

1 for=qkT m whereis equal to the standard deviation of the x and y-velocity Gaussian distribution. Hence it is useful to express the Boltzmann distribution function of speed in this form. Finally, the general form of a Boltzmann density function p(v) = v ev22(9) will be used for non-linear tting of the simulation data. It will be shown that= 1,= 2 and = 1, matching the analytically derived Boltzmann density function in 2 dimensions. The plots below refer to the x and y-velocity histograms and their corresponding Gaussian t. The last plot corresponds to the Boltzmann distribution histogram. The yellow curve is the 3D t, the red curve is the analytical t, while the blue curve is the non-linear t with least-square determined= 1:00926482,= 2.03568257 and = 1.0649113.Figure 1: Normalized distribution of x-velocities of 2500 particles 2 Figure 2: Normalized distribution of y-velocities of 2500 particles Figure 3: Normalized distribution of speeds of 2500 particles 3quotesdbs_dbs8.pdfusesText_14