Maxwell speed distribution - Revision history

Carl McBride: Changed the word velocity for speed

2012-09-17T10:40:30Z

Changed the word velocity for speed

← Older revision		Revision as of 12:40, 17 September 2012
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	The '''Maxwellian ~~velocity~~ distribution''' <ref> [http://books.google.com/books?id=hYIBOMxIuvEC&source=gbs_similarbooks_r&cad=2 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)]</ref> provides probability that the speed of a molecule of mass ''m'' lies in the range ''v'' to ''v+dv'' is given by		The '''Maxwellian speed distribution''' <ref> [http://books.google.com/books?id=hYIBOMxIuvEC&source=gbs_similarbooks_r&cad=2 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)]</ref> provides probability that the speed of a molecule of mass ''m'' lies in the range ''v'' to ''v+dv'' is given by

	:<math>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) </math>		:<math>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) </math>

Carl McBride: moved Maxwell velocity distribution to Maxwell speed distribution: Speed rather than velocity.

2012-09-17T10:40:04Z

moved Maxwell velocity distribution to Maxwell speed distribution: Speed rather than velocity.

← Older revision	Revision as of 12:40, 17 September 2012
(No difference)

Carl McBride: Updated reference with published article

2012-02-15T11:11:27Z

Updated reference with published article

← Older revision		Revision as of 13:11, 15 February 2012
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	==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 of 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 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>\mathcal T </math>. This is:		According to the '''Shivanian and Lopez-Ruiz model''' <ref>[http://dx.doi.org/10.1016/j.physa.2011.12.041 Elyas Shivanian and Ricardo López-Ruiz "A new model for ideal gases. Decay to the Maxwellian distribution", Physica A: Statistical Mechanics and its Applications '''391''' pp. 2600-2607 (2012)]
			[http://arxiv.org/abs/1105.4813 arXiv:1105.4813v1 24 May (2011)]</ref>, consider an [[ideal gas]] composed of 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 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>\mathcal T </math>. This is:

Rilopez at 10:27, 15 February 2012

2012-02-15T10:27:12Z

← Older revision		Revision as of 12:27, 15 February 2012
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	and the root-mean-square speed by		and the root-mean-square speed by

	:<math>\sqrt{\overline{v^2}} = \sqrt \frac{3}{2} v_{\rm max}</math>		:<math>\sqrt{\overline{v^2}} = \sqrt {\frac{3}{2}} v_{\rm max}</math>

Rilopez: /* Derivation */

2011-07-21T11:04:46Z

Derivation

← Older revision		Revision as of 13:04, 21 July 2011
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	==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 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>\mathcal T </math>. This is:		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 of 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 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>\mathcal T </math>. This is:

Carl McBride: /* Conjecture */ Added an internal link

2011-07-20T16:06:54Z

Conjecture: Added an internal link

← Older revision		Revision as of 18:06, 20 July 2011
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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>. 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>.		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===

Rilopez: /* Derivation */

2011-07-20T15:26:51Z

Derivation

← Older revision		Revision as of 17:26, 20 July 2011
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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===

Rilopez: /* Derivation */

2011-07-20T14:39:03Z

Derivation

← Older revision		Revision as of 16:39, 20 July 2011
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 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 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>\mathcal T </math>. This is:
	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:


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

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	===Theorem 1===		===Theorem 1===

	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>\|\|\mathcal Tp\|\|=\|\|p\|\|</math>.

	This can be interpreted as the conservation of the number of particles, or in an equivalent way, 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.
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	===Theorem 2===		===Theorem 2===

	The mean value of the velocity in the recursion <math>p_n=T^np_0</math> is conserved in time.		The mean value of the velocity in the recursion <math>p_n=\mathcal T^np_0</math> is conserved in time.
	In fact, it is null for all <math>n</math>:		In fact, it is null for all <math>n</math>:

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

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	</math>		</math>

	It means that the zero total momentum of the gas is conserved in its time evolution under the action of <math>T</math>.		It means that the zero total momentum of the gas is conserved in its time evolution under the action of <math>\mathcal T</math>.

	===Theorem 3===		===Theorem 3===
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	:<math>		:<math>
	\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 \,.		\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 \,.
	</math>		</math>

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

	===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> \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:		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_BT</math>, with <math>k_B</math> the Boltzmann constant and <math>T</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>
	\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}.		\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}.
	</math>		</math>

Carl McBride: Trivial typesetting changes.

2011-07-20T09:33:00Z

Trivial typesetting changes.

Show changes

Rilopez at 17:12, 19 July 2011

2011-07-19T17:12:30Z

← Older revision		Revision as of 19:12, 19 July 2011
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	The '''~~Maxwell~~ velocity distribution''' ~~<ref>J. C. Maxwell "", British Association for the Advancement of Science '''29''' Notices and Abstracts 9 (1859)</ref>~~		The '''Maxwellian velocity distribution''' <ref> [http://books.google.com/books?id=hYIBOMxIuvEC&source=gbs_similarbooks_r&cad=2 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)]</ref> provides probability that the speed of a molecule of mass ''m'' lies in the range ''v'' to ''v+dv'' is given by
	<ref>[http://~~dx.doi~~.~~org/10~~.~~1080~~/~~14786446008642818 J.~~ C. Maxwell "~~V. Illustrations~~ of ~~the dynamical theory of gases~~.~~—Part I~~. ~~On the motions and collisions of perfectly elastic spheres"~~, ~~Philosophical Magazine '''19''' pp. 19-32~~ ~~(1860)]</ref>~~
	~~<ref>[http://dx~~.~~doi.org/10.1080/14786446008642902 J. C. Maxwell "~~II~~. Illustrations of the dynamical theory of gases"~~, ~~Philosophical Magazine '''20''' pp. 21-37 (1860)]</ref>~~
	~~<ref>[http://dx.doi.org/10.1098/rstl.1867.0004 J. Clerk Maxwell "On the Dynamical Theory of Gases"~~, ~~Philosophical Transactions of the Royal Society of London '''157''' pp. 49-88~~ (~~1867~~)]</ref> provides probability that the speed of a molecule of mass ''m'' lies in the range ''v'' to ''v+dv'' is given by

	:<math>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) </math>		:<math>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) </math>
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	:<math>\sqrt{\overline{v^2}} = \sqrt \frac{3}{2} v_{\rm max}</math>		:<math>\sqrt{\overline{v^2}} = \sqrt \frac{3}{2} v_{\rm max}</math>


	==Derivation==		==Derivation==


	~~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		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
	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:
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	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}.		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}.
	</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>.

	==References==		==References==
	<references/>		<references/>

	;Related reading		;Related reading
	*[http://dx.~~doi~~.~~org~~/~~10.1080~~/~~002068970500044749~~ J. S. Rowlinson "The Maxwell-Boltzmann distribution", Molecular Physics '''103''' pp. 2821 - 2828 (2005)]
	*[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)]		*[http://www.ingentaconnect.com/content/tandf/tmph/2005/00000103/F0030021/art00003 J. S. Rowlinson "The Maxwell-Boltzmann distribution", Molecular Physics '''103''' pp. 2821 - 2828 (2005)]

	==External resources==		==External resources==

	*[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.24 Initial velocity distribution] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)].		*[ftp://ftp.dl.ac.uk/ccp5/ALLEN_TILDESLEY/F.24 Initial velocity distribution] sample FORTRAN computer code from the book [http://www.oup.com/uk/catalogue/?ci=9780198556459 M. P. Allen and D. J. Tildesley "Computer Simulation of Liquids", Oxford University Press (1989)].
	[[category: statistical mechanics]]		[[category: statistical mechanics]]