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Showing posts with label automata. Show all posts
Showing posts with label automata. Show all posts
02 February 2013
19 October 2012
30 April 2010
Lattice-Gas Automata for Numerical Experimental Verification of Maxwell-Boltzmann Distribution
Kontribusi Fisika Indonesia
Vol. 12 No.3, Juli 2001
68
Lattice-Gas Automata for Numerical Experimental Verification of
Maxwell-Boltzmann Distribution
Siti Nurul Khotimah, Idam Arif, and The Houw Liong
Department of Physics, Institut Teknologi Bandung
Jl. Ganesa 10 Bandung 40132
e-mail: nurul@fi.itb.ac.id
Abstract
Lattice-gas automata model has been applied to simulate the distribution function of gas molecules. This study shows a transition of a single-component velocity distribution from its initial non-equilibrium to its final equilibrium. The distribution is independent of time when the system reaches its equilibrium. For a
sufficiently dilute gas in equilibrium, the distribution function of x-velocity component is a Maxwell-Boltzmann distribution with its average velocity component is between zero and 3% of its maximum value.
This numerical experiment also obtained that the speed distribution for two-dimensional problem is a Maxwell-Boltzmann distribution. From 12 trials, average and root mean square speeds are (8.6±0.3) and (9.7±0.3) lattice units per time step respectively. We introduce a factor β to converse the unit of speed to be in meter per second. Therefore, the absolute temperature (in Kelvin) of the experiment is expressed in the mass of one molecule and Boltzmann constant as (m/k)(47.1 ± 3.2)β^2.
Keywords: Lattice-gas automata, Maxwell-Boltzmann distribution, Kinetic theory, Boltzmann transport equation
Abstrak
Model lattice-gas automata telah digunakan untuk mensimulasi fungsi distribusi molekul-molekul gas.
Makalah ini memperlihatkan suatu transisi distribusi kecepatan untuk satu komponen tertentu dari keadaan awal yang tidak setimbang menuju keaadaan akhir yang setimbang. Ketika sistem mencapai kesetimbangan,fungsi distribusinya tidak bergantung pada waktu. Untuk gas yang kerapatan molekulnya cukup rendah dan
berada dalam kesetimbangan, fungsi distribusi untuk komponen kecepatan dalam arah-x berupa distribusi Maxwell-Boltzmann dengan nilai rata-ratanya antara nol sampai 3% nilai maksiumunya. Percobaan numerik ini juga mendapatkan hasil bahwa distribusi laju untuk kasus dua-dimensi adalah berupa distribusi Maxwell-Boltzmann. Dari 12 pengulangan percobaan, laju rata-rata dan laju akar-rata-rata-kuadratnya
adalah (8.6±0.3) dan (9.7±0.3) satuan kisi per satuan waktu. Kami memperkenalkan faktor β untuk mengubah satuan laju menjadi bersatuan meter per sekon. Oleh karena itu, temperatur mutlak (dalam satuan Kelvin) pada percobaan ini dinyatakan dalam massa sebuah molekul dan konstanta Boltzmann sebagai berikut: (m/k)(47.1 ± 3.2)β^2.
Kata kunci: Lattice-gas automata, Distribusi Maxwell-Boltzmann, Teori kinetik, Persamaan transport
Boltzmann
Vol. 12 No.3, Juli 2001
68
Lattice-Gas Automata for Numerical Experimental Verification of
Maxwell-Boltzmann Distribution
Siti Nurul Khotimah, Idam Arif, and The Houw Liong
Department of Physics, Institut Teknologi Bandung
Jl. Ganesa 10 Bandung 40132
e-mail: nurul@fi.itb.ac.id
Abstract
Lattice-gas automata model has been applied to simulate the distribution function of gas molecules. This study shows a transition of a single-component velocity distribution from its initial non-equilibrium to its final equilibrium. The distribution is independent of time when the system reaches its equilibrium. For a
sufficiently dilute gas in equilibrium, the distribution function of x-velocity component is a Maxwell-Boltzmann distribution with its average velocity component is between zero and 3% of its maximum value.
This numerical experiment also obtained that the speed distribution for two-dimensional problem is a Maxwell-Boltzmann distribution. From 12 trials, average and root mean square speeds are (8.6±0.3) and (9.7±0.3) lattice units per time step respectively. We introduce a factor β to converse the unit of speed to be in meter per second. Therefore, the absolute temperature (in Kelvin) of the experiment is expressed in the mass of one molecule and Boltzmann constant as (m/k)(47.1 ± 3.2)β^2.
Keywords: Lattice-gas automata, Maxwell-Boltzmann distribution, Kinetic theory, Boltzmann transport equation
Abstrak
Model lattice-gas automata telah digunakan untuk mensimulasi fungsi distribusi molekul-molekul gas.
Makalah ini memperlihatkan suatu transisi distribusi kecepatan untuk satu komponen tertentu dari keadaan awal yang tidak setimbang menuju keaadaan akhir yang setimbang. Ketika sistem mencapai kesetimbangan,fungsi distribusinya tidak bergantung pada waktu. Untuk gas yang kerapatan molekulnya cukup rendah dan
berada dalam kesetimbangan, fungsi distribusi untuk komponen kecepatan dalam arah-x berupa distribusi Maxwell-Boltzmann dengan nilai rata-ratanya antara nol sampai 3% nilai maksiumunya. Percobaan numerik ini juga mendapatkan hasil bahwa distribusi laju untuk kasus dua-dimensi adalah berupa distribusi Maxwell-Boltzmann. Dari 12 pengulangan percobaan, laju rata-rata dan laju akar-rata-rata-kuadratnya
adalah (8.6±0.3) dan (9.7±0.3) satuan kisi per satuan waktu. Kami memperkenalkan faktor β untuk mengubah satuan laju menjadi bersatuan meter per sekon. Oleh karena itu, temperatur mutlak (dalam satuan Kelvin) pada percobaan ini dinyatakan dalam massa sebuah molekul dan konstanta Boltzmann sebagai berikut: (m/k)(47.1 ± 3.2)β^2.
Kata kunci: Lattice-gas automata, Distribusi Maxwell-Boltzmann, Teori kinetik, Persamaan transport
Boltzmann
26 April 2010
LATTICE-GAS AUTOMATA FOR THE PROBLEM OF KINETIC THEORY OF GAS DURING FREE EXPANSION
International Journal of Modern Physics C (IJMPC)
Computational Physics and Physical Computation
Current Issue | 2010 | 2009 | 2008 | All Volumes (1990-2010)
Volume: 13, Issue: 8(2002) pp. 1033-1045 DOI: 10.1142/S0129183102003772
Abstract | Full Text (PDF, 1,800KB)
Title: LATTICE-GAS AUTOMATA FOR THE PROBLEM OF KINETIC THEORY OF GAS DURING FREE EXPANSION
Author(s):
SITI NURUL KHOTIMAH
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
IDAM ARIF
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
THE HOUW LIONG
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
History:
Received 27 September 2001
Revised 25 April 2002
Abstract:
The lattice-gas method has been applied to solve the problem of kinetic theory of gas in the Gay–Lussac–Joule experiment. Numerical experiments for a two-dimensional gas were carried out to determine the number of molecules in one vessel (Nr), the ratio between the mean square values of the components of molecule velocity , and the change in internal energy (ΔU) as a function of time during free expansion. These experiments were repeated for different sizes of an aperture in the partition between the two vessels.
After puncturing the partition, the curve for the particle number in one vessel shows a damped oscillation for about half of the total number. The oscillations do not vanish after a sampling over different initial configurations. The system is in nonequilibrium due to the pressure equilibration, and here the flow is actually compressible. The equilibration time (in time steps) decreases with decreased size of aperture in the partition. For very small apertures (equal or less than g.3^0.5/2 lattice units), the number of molecules in one vessel changes with time in a smooth way until it reaches half of the total number; their curves obey the analytical solution for quasi-static processes. The calculations on v_x^2/v_y^2 and ΔU also support the results that the equilibration time decreases with decreased size of aperture in the partition.
Keywords:
Lattice-gas automata; kinetic theory; Boltzmann transport equation; free expansion; Gay–Lussac–Joule experiment
Computational Physics and Physical Computation
Current Issue | 2010 | 2009 | 2008 | All Volumes (1990-2010)
Volume: 13, Issue: 8(2002) pp. 1033-1045 DOI: 10.1142/S0129183102003772
Abstract | Full Text (PDF, 1,800KB)
Title: LATTICE-GAS AUTOMATA FOR THE PROBLEM OF KINETIC THEORY OF GAS DURING FREE EXPANSION
Author(s):
SITI NURUL KHOTIMAH
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
IDAM ARIF
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
THE HOUW LIONG
Department of Physics, Institut Teknologi Bandung, jl. Ganesha 10 Bandung 40132, Indonesia
History:
Received 27 September 2001
Revised 25 April 2002
Abstract:
The lattice-gas method has been applied to solve the problem of kinetic theory of gas in the Gay–Lussac–Joule experiment. Numerical experiments for a two-dimensional gas were carried out to determine the number of molecules in one vessel (Nr), the ratio between the mean square values of the components of molecule velocity , and the change in internal energy (ΔU) as a function of time during free expansion. These experiments were repeated for different sizes of an aperture in the partition between the two vessels.
After puncturing the partition, the curve for the particle number in one vessel shows a damped oscillation for about half of the total number. The oscillations do not vanish after a sampling over different initial configurations. The system is in nonequilibrium due to the pressure equilibration, and here the flow is actually compressible. The equilibration time (in time steps) decreases with decreased size of aperture in the partition. For very small apertures (equal or less than g.3^0.5/2 lattice units), the number of molecules in one vessel changes with time in a smooth way until it reaches half of the total number; their curves obey the analytical solution for quasi-static processes. The calculations on v_x^2/v_y^2 and ΔU also support the results that the equilibration time decreases with decreased size of aperture in the partition.
Keywords:
Lattice-gas automata; kinetic theory; Boltzmann transport equation; free expansion; Gay–Lussac–Joule experiment
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