Maxwell speed distribution: Difference between revisions
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The one-parametric family of normalized Gaussian functions <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}e^{-\alpha x^2}</math>, <math>\alpha\ge 0</math>, <math>||p_{\alpha}||=1</math>, are fixed points of the operator <math>\mathcal T</math>. In other words, <math>\mathcal Tp_{\alpha}=p_{\alpha}</math>. | The one-parametric family of normalized Gaussian functions <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}e^{-\alpha x^2}</math>, <math>\alpha\ge 0</math>, <math>||p_{\alpha}||=1</math>, are fixed points of the operator <math>\mathcal T</math>. In other words, <math>\mathcal Tp_{\alpha}=p_{\alpha}</math>. | ||
===Theorem 5=== | |||
For distributions <math>p</math> with <math>||p||=1</math>, suppose that <math>\lim_{n\rightarrow\infty}||\mathcal T^np(x)-\mu(x)||=0</math>, and <math>\mu(x)</math> is a normalized continuous distribution, then <math>\mu(x)</math> is a fixed point of the operator <math>\mathcal T </math>. | |||
===Conjecture=== | ===Conjecture=== | ||
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As a consequence of the former theorems, and by simulation of many examples, the following conjecture can be stated: | As a consequence of the former theorems, and by simulation of many examples, the following conjecture can be stated: | ||
For any <math>p</math> with <math>||p||=1</math>, with finite <math> \langle x^2,p \rangle </math> and verifying <math>\lim_{n\rightarrow\infty} ||\mathcal T^np(x)-\mu(x)||=0</math>, the limit <math>\mu(x)</math> is the fixed point <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}\,e^{-\alpha x^2}</math>, with <math>\alpha=(2\, \langle x^2,p \rangle)^{-1}</math>. | For any <math>p</math> with <math>||p||=1</math>, with finite <math> \langle x^2,p \rangle </math> and verifying <math>\lim_{n\rightarrow\infty} ||\mathcal T^np(x)-\mu(x)||=0</math>, the limit <math>\mu(x)</math> is the fixed point <math>p_{\alpha}(x)=\sqrt{\alpha\over\pi}\,e^{-\alpha x^2}</math>, with <math>\alpha=(2\, \langle x^2,p \rangle)^{-1}</math>. That is, the asymptotic steady state is the gaussian distribution with the same mean energy than the initial out-of-equilibrium state <math>p</math>. | ||
===Conclusion=== | ===Conclusion=== | ||
Revision as of 16:26, 20 July 2011
The Maxwellian velocity distribution [1] provides probability that the speed of a molecule of mass m lies in the range v to v+dv is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(v)dv = 4 \pi v^2 dv \left( \frac{m}{2 \pi k_B T} \right)^{3/2} \exp (-mv^2/2k_B T) }
where T is the temperature and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant. The maximum of this distribution is located at
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_{\rm max} = \sqrt{\frac{2k_BT}{m}}}
The mean speed is given by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{v} = \frac{2}{\sqrt \pi} v_{\rm max}}
and the root-mean-square speed by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\overline{v^2}} = \sqrt \frac{3}{2} v_{\rm max}}
Derivation
According to the Shivanian and Lopez-Ruiz model [2], consider an ideal gas composed particles having a mass of unity in the three-dimensional (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3D} ) space. As long as there no privileged direction when in equilibrium, we can take any direction in space and study the discrete time evolution of the velocity distribution in that direction. Let us call this axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} . We can complete a Cartesian system with two additional orthogonal axis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V,W} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(u){\mathrm d}u} represents the probability of finding a particle of the gas with velocity component in the direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} comprised between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u + {\mathrm d}u} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , then the probability to have at this time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} a particle with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3D} velocity will be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(u)p_n(v)p_n(w)} . The particles of the gas collide between them, and after a number of interactions of the order of system size, a new velocity distribution is attained at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} . Concerning the interaction of particles with the bulk of the gas, we make two simplistic and realistic assumptions in order to obtain the probability of having a velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} : (1) Only those particles with an energy greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} can contribute to this velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the direction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} , that is, all those particles whose velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v,w)} verify ; (2) The new velocities after collisions are equally distributed in their permitted ranges, that is, particles with velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v,w)} can generate maximal velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm U_{max}=\pm\sqrt{u^2+v^2+w^2}} , then the allowed range of velocities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-U_{max},U_{max}]} measures Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2|U_{max}|} , and the contributing probability of these particles to the velocity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} will be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(u)p_n(v)p_n(w)/(2|U_{max}|)} . Taking all together we finally get the expression for the evolution operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T } . This is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}(x)=\mathcal Tp_n(x) = \iiint_{u^2+v^2+w^2\ge x^2}\,{p_n(u)p_n(v)p_n(w)\over 2\sqrt{u^2+v^2+w^2}} \; {\mathrm d}u~{\mathrm d}v~{\mathrm d}w\,. }
Let us remark that we have not made any supposition about the type of interactions or collisions between the particles and, in some way, the equivalent of the Boltzmann hypothesis of molecular chaos would be the two simplistic assumptions we have stated on the interaction of particles with the bulk of the gas. In fact, the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} conserves the energy and the null momentum of the gas over time. Moreover, for any initial velocity distribution, the system tends towards its equilibrium, i.e. towards the Maxwellian Velocity Distribution (MVD). This means that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow\infty} \mathcal T^n \left(p_0(x)\right) \rightarrow p_f(x)= \mathrm{MVD}\;(1D\;case)\,. }
Let us sketch now all these properties. First, we introduce the norm of positive functions (one-dimensional velocity distributions) in the real axis as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vert\vert p\vert\vert=\int_{-\infty}^{+\infty} p(x) dx. }
Then we have the following exact results:
Theorem 1
For any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||p||=1} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||\mathcal Tp||=||p||} .
This can be interpreted as the conservation of the number of particles, or in an equivalent way, the total mass of the gas.
Theorem 2
The mean value of the velocity in the recursion is conserved in time. In fact, it is null for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x,\mathcal Tp \rangle = \langle x,\mathcal T^2p \rangle = \langle x,\mathcal T^3p \rangle=\cdots= \langle x,\mathcal T^np \rangle =\cdots=0\,, }
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle f,g \rangle =\int_{-\infty}^{+\infty}f(x)g(x){\mathrm d}x\,. }
It means that the zero total momentum of the gas is conserved in its time evolution under the action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} .
Theorem 3
For every with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||p||=1} , we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x^2,p \rangle= \langle x^2,\mathcal Tp \rangle= \langle x^2,\mathcal T^2p \rangle= \langle x^2,\mathcal T^3p \rangle =\cdots= \langle x^2,\mathcal T^np \rangle=\cdots \,. }
It means that the mean energy per particle is conserved and in consequence, by Theorem 1, the total energy of the gas is conserved in time.
Theorem 4
The one-parametric family of normalized Gaussian functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{\alpha}(x)=\sqrt{\alpha\over\pi}e^{-\alpha x^2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha\ge 0} , , are fixed points of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T} . In other words, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal Tp_{\alpha}=p_{\alpha}} .
Theorem 5
For distributions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ||p||=1} , suppose that , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(x)} is a normalized continuous distribution, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(x)} is a fixed point of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal T } .
Conjecture
As a consequence of the former theorems, and by simulation of many examples, the following conjecture can be stated:
For any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} with , with finite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x^2,p \rangle } and verifying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n\rightarrow\infty} ||\mathcal T^np(x)-\mu(x)||=0} , the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(x)} is the fixed point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{\alpha}(x)=\sqrt{\alpha\over\pi}\,e^{-\alpha x^2}} , with . That is, the asymptotic steady state is the gaussian distribution with the same mean energy than the initial out-of-equilibrium state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} .
Conclusion
In physical terms, it means that for any initial velocity distribution of the gas, it decays to the Maxwellian distribution, which is just the fixed point of the dynamics. Recalling that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x^2,p \rangle=k_BT} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} the Boltzmann constant and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} the temperature of the gas, and introducing the mass of the particles, let us observe that the MVD (above presented) is recovered in its Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3D} format:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{MVD} = p_{\alpha}(u)p_{\alpha}(v)p_{\alpha}(w)=\left({m\alpha\over\pi}\right)^{3\over 2}\,\exp^{-m\alpha (u^2+v^2+w^2)} \;\;\; with \;\;\; \alpha=(2k_B T)^{-1}. }
Moreover, an increase in the entropy is found during all the decay process. This gives rise to the celebrated H-theorem [3].
References
- ↑ James C. Maxwell, "The scientific papers of James Clerk Maxwell", Edited by W.D. Niven, paper number XX, Dover Publications, Vol. I,II, New York, USA (2003)
- ↑ Elyas Shivanian and Ricardo Lopez-Ruiz "A New Model for Ideal Gases. Decay to the Maxwellian Distribution", arXiv:1105.4813v1 24 May (2011)
- ↑ Ludwig Boltzmann, "Lectures on Gas Theory", Translated by S.G. Brush, Dover Publications, New York, USA (1995) ISBN 0486684555
- Related reading
External resources
- Initial velocity distribution sample FORTRAN computer code from the book M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989).