Maxwell speed distribution: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
No edit summary
m (Trivial typesetting changes.)
Line 18: Line 18:


==Derivation==
==Derivation==
 
According to the '''Shivanian and Lopez-Ruiz model''' <ref>[http://arxiv.org/abs/1105.4813 Elyas Shivanian and Ricardo Lopez-Ruiz "A New Model for Ideal Gases. Decay to the Maxwellian Distribution", arXiv:1105.4813v1 24 May (2011)]</ref>, consider an [[ideal gas]] composed particles having a mass of unity in the three-dimensional (<math>3D</math>) 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 <math>U</math>. We can complete a Cartesian system with two additional orthogonal axis <math>V,W</math>. If <math>p_n(u){\mathrm d}u</math> represents the probability of finding a particle of the gas with velocity component in the direction <math>U</math> comprised between <math>u</math> and <math>u + {\mathrm d}u</math> at time <math>n</math>, then the probability to have at this time <math>n</math> a particle with a <math>3D</math> velocity <math>(u,v,w)</math> will be <math>p_n(u)p_n(v)p_n(w)</math>. 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 <math>n+1</math>. 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 <math>x</math> in the direction <math>U</math> at time <math>n+1</math>: (1) Only those particles with an energy greater than <math>x^2</math> at time <math>n</math> can contribute to this velocity <math>x</math> in the direction <math>U</math>, that is, all those particles whose velocities <math>(u,v,w)</math> verify <math> u^2+v^2+w^2\ge x^2</math>; (2) The new velocities after collisions are equally  
 
According to the '''Shivanian & Lopez-Ruiz model''' <ref>[http://arxiv.org/abs/1105.4813 Elyas Shivanian and Ricardo Lopez-Ruiz "A New Model for Ideal Gases. Decay to the Maxwellian Distribution", arXiv:1105.4813v1 24 May (2011)]</ref>, consider an ideal gas with particles of unity mass in the three-dimensional (<math>3D</math>) space. As long as there is not a privileged direction in the equilibrium, we can take any direction in the space and to study the discrete time evolution of the velocity distribution in that direction. Let us call this axis <math>U</math>. We can complete a Cartesian system with two additional orthogonal axis <math>V,W</math>. If <math>p_n(u){\mathrm d}u</math> represents the probability of finding a particle of the gas with velocity component in the direction <math>U</math> comprised between <math>u</math> and <math>u + {\mathrm d}u</math> at time <math>n</math>, then the probability to have at this time <math>n</math> a particle with a <math>3D</math> velocity <math>(u,v,w)</math> will be <math>p_n(u)p_n(v)p_n(w)</math>. 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 <math>n+1</math>. 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 <math>x</math> in the direction <math>U</math> at time <math>n+1</math>: (1) Only those particles with an energy bigger than <math>x^2</math> at time <math>n</math> can contribute to this velocity <math>x</math> in the direction <math>U</math>, that is, all those particles whose velocities <math>(u,v,w)</math> verify <math> u^2+v^2+w^2\ge x^2</math>; (2) The new velocities after collisions are equally  
distributed in their permitted ranges, that is, particles with velocity <math>(u,v,w)</math> can generate maximal velocities <math>\pm U_{max}=\pm\sqrt{u^2+v^2+w^2}</math>, then the allowed range of velocities <math>[-U_{max},U_{max}]</math> measures <math>2|U_{max}|</math>, and the contributing probability of these particles to the velocity <math>x</math> will be <math>p_n(u)p_n(v)p_n(w)/(2|U_{max}|)</math>. Taking all together we finally get the expression  
distributed in their permitted ranges, that is, particles with velocity <math>(u,v,w)</math> can generate maximal velocities <math>\pm U_{max}=\pm\sqrt{u^2+v^2+w^2}</math>, then the allowed range of velocities <math>[-U_{max},U_{max}]</math> measures <math>2|U_{max}|</math>, and the contributing probability of these particles to the velocity <math>x</math> will be <math>p_n(u)p_n(v)p_n(w)/(2|U_{max}|)</math>. Taking all together we finally get the expression  
for the evolution operator <math>T</math>. This is:
for the evolution operator <math>T</math>. This is:




<math>
:<math>
p_{n+1}(x)=Tp_n(x) = \int\int\int_{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\,.
p_{n+1}(x)=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\,.
</math>
</math>


Let us remark that we have not made any supposition about the type of interactions or collisions
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
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 <math>T</math> conserves in time the energy and the null momentum of the gas. Moreover, for any initial velocity distribution, the system tends towards its equilibrium, i.e. towards the Maxwellian Velocity Distribution (MVD). This means that
the bulk of the gas. In fact, the operator <math>T</math> 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
 


<math>
:<math>
\lim_{n\rightarrow\infty} T^n \left(p_0(x)\right) \rightarrow p_f(x)=MVD\;(1D\;case)\,.
\lim_{n\rightarrow\infty} T^n \left(p_0(x)\right) \rightarrow p_f(x)= \mathrm{MVD}\;(1D\;case)\,.
</math>
</math>


Let us sketch now all these properties.


First, we introduce the norm <math>||\cdot||</math> of positive functions (one-dimensional velocity distributions) in the real axis as
Let us sketch now all these properties. First, we introduce the norm <math>||\cdot||</math> of positive functions (one-dimensional velocity distributions) in the real axis as


<math>
:<math>
\vert\vert p\vert\vert=\int_{-\infty}^{+\infty} p(x) dx.
\vert\vert p\vert\vert=\int_{-\infty}^{+\infty} p(x) dx.
</math>
</math>
Line 51: Line 49:
For any <math>p</math> with <math>||p||=1</math>, we have <math>||Tp||=||p||</math>.
For any <math>p</math> with <math>||p||=1</math>, we have <math>||Tp||=||p||</math>.


This can be interpreted as the conservation of the number of particles or in an equivalet way of the total mass of the gas.
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===
===Theorem 2===
Line 58: Line 56:
In fact, it is null for all <math>n</math>:
In fact, it is null for all <math>n</math>:


<math>
:<math>
<x,Tp>=<x,T^2p>=<x,T^3p>=\cdots=<x,T^np>=\cdots=0\,,
\langle x,Tp \rangle = \langle x,T^2p \rangle = \langle x,T^3p \rangle=\cdots= \langle x,T^np \rangle =\cdots=0\,,
</math>
</math>


where
where


<math>
:<math>
<f,g>=\int_{-\infty}^{+\infty}f(x)g(x){\mathrm d}x\,.
\langle f,g \rangle =\int_{-\infty}^{+\infty}f(x)g(x){\mathrm d}x\,.
</math>
</math>


Line 74: Line 72:
For every <math>p</math> with <math>||p||=1</math>, we have
For every <math>p</math> with <math>||p||=1</math>, we have


<math>
:<math>
<x^2,p>=<x^2,Tp>=<x^2,T^2p>=<x^2,T^3p>=\cdots=<x^2,T^np>=\cdots \,.
\langle x^2,p \rangle= \langle x^2,Tp \rangle= \langle x^2,T^2p \rangle= \langle x^2,T^3p \rangle =\cdots= \langle x^2,T^np \rangle=\cdots \,.
</math>
</math>


Line 82: Line 80:
===Theorem 4===
===Theorem 4===


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>T</math>. In other words, <math>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>T</math>. In other words, <math>Tp_{\alpha}=p_{\alpha}</math>.


===Conjecture===
===Conjecture===


As a consequence of the former theorems and by simulation of many exmaples, it can be claimed the following conjecture:
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><x^2,p></math> and verifying <math>\lim_{n\rightarrow\infty} ||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\,<x^2,p>)^{-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} ||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>.  


===Conclusion===
===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 <math><x^2,p>=k\tau</math>, with <math>k</math> the Boltzmann constant and <math>\tau</math> the temperature of the gas, and introducing the mass <math>m</math> of the particles, let us observe that the MVD (above presented) is recovered in its <math>3D</math> format:  
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 <math> \langle x^2,p \rangle=k\tau</math>, with <math>k</math> the Boltzmann constant and <math>\tau</math> the temperature of the gas, and introducing the mass <math>m</math> of the particles, let us observe that the MVD (above presented) is recovered in its <math>3D</math> format:  


<math>
:<math>
MVD = p_{\alpha}(u)p_{\alpha}(v)p_{\alpha}(w)=\left({m\alpha\over\pi}\right)^{3\over 2}\,e^{-m\alpha (u^2+v^2+w^2)} \;\;\; with \;\;\;  \alpha=(2k\tau)^{-1}.
\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\tau)^{-1}.
</math>
</math>


Moreover, the increasing of the entropy is found during all the decay process. This gives rise to the celebrated [[H-theorem]] <ref> [http://books.google.pn/books/about/Lectures_on_Gas_Theory.html?id=-I7QzCXnstEC Ludwig Boltzmann, "Lectures on Gas Theory", Translated by S.G. Brush, Dover Publications, New York, USA (1995)]</ref>.
Moreover, an increase in the [[entropy]] is found during all the decay process. This gives rise to the celebrated [[H-theorem]] <ref>Ludwig Boltzmann, "Lectures on Gas Theory", Translated by S.G. Brush, Dover Publications, New York, USA (1995) ISBN 0486684555</ref>.


==References==
==References==

Revision as of 10:33, 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 (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 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)} 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 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 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^2+v^2+w^2\ge x^2} ; (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 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)=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 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} 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 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 ||\cdot||} 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 ||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 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=T^np_0} 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,Tp \rangle = \langle x,T^2p \rangle = \langle x,T^3p \rangle=\cdots= \langle x,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 T} .

Theorem 3

For every 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 \langle x^2,p \rangle= \langle x^2,Tp \rangle= \langle x^2,T^2p \rangle= \langle x^2,T^3p \rangle =\cdots= \langle x^2,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} , 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}||=1} , 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 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 Tp_{\alpha}=p_{\alpha}} .

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 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} , 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} ||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 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=(2\, \langle x^2,p \rangle)^{-1}} .

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\tau} , 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} 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 \tau} the temperature of the gas, and introducing the mass 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 m} 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\tau)^{-1}. }

Moreover, an increase in the entropy is found during all the decay process. This gives rise to the celebrated H-theorem [3].

References

  1. ↑ 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)
  2. ↑ Elyas Shivanian and Ricardo Lopez-Ruiz "A New Model for Ideal Gases. Decay to the Maxwellian Distribution", arXiv:1105.4813v1 24 May (2011)
  3. ↑ Ludwig Boltzmann, "Lectures on Gas Theory", Translated by S.G. Brush, Dover Publications, New York, USA (1995) ISBN 0486684555
Related reading

External resources

Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php?title=Maxwell_speed_distribution&oldid=11616"