Gallery :

Statistical Mechanics and Particle Systems

[Galerie : Mécanique Statistique et Systèmes de Particules]

• Bidimensional Random Walks [Marches Aléatoires Bidimensionnelles] :
• Tridimensional Random Walks [Marches Aléatoires Tridimensionnelles] :
• Bidimensional Billiards [Billards Bidimensionnels] :
• Perfect Impacts [Chocs Parfaits] :
• Tridimensional Billiards [Billards Tridimensionnels] :
• Bidimensional Diffusion [Diffusion Bidimensionnelle] :
• Tridimensional Diffusion [Diffusion Tridimensionnelle] :
• Bidimensional Brownian Motion [Mouvement Brownien Bidimensionnel] :
• Tridimensional Brownian Motion [Mouvement Brownien Tridimensionnel] :
• Bidimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Bidimensionnels] :
• Fractal Diffusion Front in a Bidimensional Medium [Front Fractal de Diffusion dans un Milieu Bidimensionnel] :
• Tridimensional Diffusion Limited Aggregations Fractal Aggregates [DLAs et Agrégats Fractals Tridimensionnels] :
• Bidimensional Fluids [Fluides Bidimensionnels] :
• Tridimensional Fluids [Fluides Tridimensionnels] :
• Thermodynamics [Thermodynamique] :
• Bidimensional Spin Networks [Réseaux Bidimensionnels de Spins] :
• Tridimensional Spin Networks [Réseaux Tridimensionnels de Spins] :

Statistical Mechanics and Particle Systems [Mécanique Statistique et Systèmes de Particules] :

Bidimensional Random Walks [Marches Aléatoires Bidimensionnelles] :

When displaying the gravity center of a set of particles, its trajectory can hide one or more of these particles. In this case, the written number of particles seems to be wrong : but it is not !



Isotropic random walk of 4 particles on a bidimensional square lattice with display of their gravity center -white particle-.	Isotropic random walk of 16 particles on a bidimensional square lattice with display of their gravity center -white particle-.	Isotropic random walk of 64 particles on a bidimensional square lattice with display of their gravity center -white particle-.	Isotropic random walk of 256 particles on a bidimensional square lattice with display of their gravity center -white particle-.	Isotropic random walk of 64 particles on a bidimensional square lattice.
Isotropic random walk of 64 particles on a bidimensional square lattice.	Non isotropic random walk of 64 particles on a bidimensional square lattice in a 'right-shift' potential.	Non isotropic random walk of 64 particles on a bidimensional square lattice in a 'right-shift' potential.	Non isotropic random walk of 64 particles on a bidimensional square lattice in a 'ring' potential.	Non isotropic random walk of 64 particles on a bidimensional square lattice in a 'ring' potential.

The random walk of photons escaping the Sun.




Tridimensional Random Walks [Marches Aléatoires Tridimensionnelles] :

Isotropic random walk of 64 particles on a tridimensional square lattice.

Isotropic random walk of 64 particles on a tridimensional square lattice.




Bidimensional Billiards [Billards Bidimensionnels] :

When displaying the gravity center of a set of particles, its trajectory can hide one or more of these particles. In this case, the written number of particles seems to be wrong : but it is not !



Bidimensional rectangular billiard with 64 isotropic particles and without collisions.	Bidimensional rectangular billiard with 64 isotropic particles and with collisions.	Double bidimensional rectangular billiard with 64 isotropic particles and without collisions.	Double bidimensional rectangular billiard with 64 isotropic particles and with collisions.	Bidimensional Sierpinski billiard with 64 isotropic particles and without collisions.
Bidimensional Sierpinski billiard with 64 isotropic particles and with collisions.	Bidimensional snowflake billiard with 64 isotropic particles and with collisions.	Bidimensional Mandelbrot billiard with 64 isotropic particles and without collisions.	Bidimensional Mandelbrot billiard with 64 isotropic particles and with collisions.	Bidimensional rectangular billiard with 4 random particles, with collisions and display of their gravity center -white particle-.
Bidimensional rectangular billiard with 16 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional rectangular billiard with 64 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional rectangular billiard with 256 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional rectangular billiard with 50% of small particles and 50% of big particles -small and big particles have the same mass-.	Bidimensional rectangular billiard with 50% of small light particles and 50% of big heavy particles.
Bidimensional rectangular billiard with 50% of small particles and 50% of big particles submitted to a vertical field of gravity -small and big particles have the same mass-.	Bidimensional rectangular billiard with 50% of small light particles and 50% of big heavy particles submitted to a vertical field of gravity.	Bidimensional billiard with 6 still -initially- particles submitted to an attractive central field of gravity.	Bidimensional billiard with 6 still -initially- particles submitted to an attractive central field of gravity (the center being the magenta lower-left particle).

Artistic view of the trajectories of 12 -initially- still particles submitted to an attractive central field of gravity in a tridimensional billiard.

Bidimensional rectangular billiard with a central obstacle and an initial flow of rotating particles.	Bidimensional crenellated billiard with 678 isotropic interacting particles.	Bidimensional crenellated billiard with 678 isotropic non interacting particles.	Bidimensional disymmetrical double billiard with 470 isotropic non interacting particles.	Bidimensional disymmetrical double billiard with 321 isotropic non interacting particles.
Bidimensional snowflake-like billiard with 391 isotropic non interacting particles.	Bidimensional Sierpinski billiard with 4 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional Sierpinski billiard with 16 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional Sierpinski billiard with 64 random particles, with collisions and display of their gravity center -white particle-.	Bidimensional Sierpinski billiard with 256 random particles, with collisions and display of their gravity center -white particle-.
Bidimensional Sierpinski billiard with 562 isotropic particles submitted to a vertical field of gravity.	Bidimensional Sierpinski billiard with 326 isotropic particles submitted to a strong vertical field of gravity.	The irreversibility of the numerical bidimensional billiard.	Bidimensional domino effect.

Time-dependent billiard (from negative curvatures to positive curvature) with one particle.

Time-dependent billiard (from negative curvatures to positive curvature) with one accelerated particle.

Time-dependent billiard (from negative curvatures to positive curvature) with one accelerated particle.




Perfect Impacts [Chocs Parfaits] :

Perfect impact between 2 particles (yellow and blue, white point is their gravity center).

Perfect impact between 2 particles (yellow and blue) -display with respect of the gravity center (white particle)-.




Tridimensional Billiards [Billards Tridimensionnels] :

Tridimensional billiard with 12 -initially- still particles submitted to an attractive central field of gravity.

Tridimensional domino effect.




Bidimensional Diffusion [Diffusion Bidimensionnelle] :

Diffusion between two boxes (initial conditions : the left one is empty, the right one contains 256 particles), with collisions and display of the gravity center -white particle-.

Diffusion between two boxes (initial conditions : the left one is empty, the right one contains 256 particles), with collisions and a time-dependent geometry.

Spreading of an epidemic -color changes by means of contacts-.

Diffusion between two boxes by means of an osmotic process.




Tridimensional Diffusion [Diffusion Tridimensionnelle] :

Particle diffusion inside the Hiroko Kitaoka model of the human pulmonary acinus.	Particle diffusion inside the Hiroko Kitaoka model of the human pulmonary acinus.	Particle diffusion inside the Hiroko Kitaoka model of the human pulmonary acinus with membrane permeability.	Particle diffusion inside the Hiroko Kitaoka model of the human pulmonary acinus with membrane permeability.	A gradient of 114299 particles obtains by means of a tridimensional diffusion process.
A gradient of 128312 particles obtains by means of a tridimensional diffusion process.




Bidimensional Brownian Motion [Mouvement Brownien Bidimensionnel] :

Brownian motion of a few heavy slow particles inside a gaz of fast light particles.	Brownian motion of a few heavy slow particles inside a gaz of fast light particles.	Brownian motion of a few heavy slow particles inside a gaz of fast light particles with increasing mass ratio (heavy/light=1,10,100,1000).	Bidimensional brownian motion -dark green- and its 'external border' -white-.	Bidimensional brownian motion -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time- and its 'external border' -white-.
Bidimensional brownian motion -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time- and its 'external border' -white-.	Bidimensional brownian motion with a uniform distribution law regarding the radial coordinate.	Bidimensional brownian motion with a gaussian distribution law regarding the radial coordinate.	Bidimensional brownian motion with a gaussian-like (with two bells) distribution law regarding the radial coordinate.	Bidimensional brownian motion with three different distribution laws regarding the radial coordinate -20 points starting from the white point at the picture center-.
Bidimensional brownian motion with three different distribution laws regarding the radial coordinate -60 points starting from the white point at the picture center-.	Bidimensional brownian motion on a square lattice based on a random process -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time-.	Bidimensional brownian motion on a square lattice based on the Verhulst dynamics -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time-.




Tridimensional Brownian Motion [Mouvement Brownien Tridimensionnel] :

Tridimensional brownian motion -dark green- and the 'external border' of its bidimensional projection -white-.

Tridimensional brownian motion -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time- and the 'external border' of its bidimensional projection -white-.

Tridimensional brownian motion -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time- and the 'external border' of its bidimensional projection -white-.

Tridimensional brownian motion on a cubic lattice based on a random process -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time-.

Tridimensional brownian motion on a cubic lattice based on the Verhulst dynamics -the colors used (magenta,red,yellow,green,cyan) are an increasing function of the time-.




Bidimensional Diffusion Limited Aggregations and Fractal Aggregates [DLAs et Agrégats Fractals Bidimensionnels] :

Bidimensional fractal aggregates obtained by means of pasting during collisions with display of the velocity histogram.	Bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	The trajectories of bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	Bidimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	Bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.
The trajectories of bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.	Bidimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.	Artistic view of the trajectories of bidimensional fractal aggregates obtained by means of pasting during collisions of particles submitted to an attractive central field of gravity.	Bidimensional quasi-symmetrical fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	Bidimensional quasi-symmetrical fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.
Bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	The trajectories of bidimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	Bidimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	Bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.	The trajectories of bidimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
Bidimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.	Bidimensional fractal aggregates obtained by means of two families of interacting particles with the following initial conditions about velocity : V for the Red ones and V/5 for the Blue ones.




Fractal Diffusion Front in a Bidimensional Medium [Front Fractal de Diffusion dans un Milieu Bidimensionnel] :

Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.

Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.

Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.

Fractal diffusion front in a bidimensional medium obtained by means of a random walk process.

Tridimensional display (bird's-eye view) of the density of particles during a bidimensional diffusion process.

Tridimensional display (bird's-eye view) of the density of particles during a bidimensional diffusion process.

Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles.

Fractal diffusion front in a bidimensional medium obtained by means of identical interacting particles with spatial Fourier filtering.




Tridimensional Diffusion Limited Aggregations Fractal Aggregates [DLAs et Agrégats Fractals Tridimensionnels] :

Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to an attractive central field of gravity.	Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.	Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to an attractive central field of gravity.	Rotation about the Y (vertical) axis of a tridimensional fractal aggregate that can also be viewed as a set of 4x3 pseudo-stereograms.
Tridimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	The trajectories of tridimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	Tridimensional fractal aggregate obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.	Tridimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.	The trajectories of tridimensional fractal aggregates obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.
Tridimensional fractal aggregate obtained by means of a 50% pasting process during collisions of particles submitted to a vertical field of gravity.	Artistic view of the trajectories of tridimensional fractal aggregates obtained by means of a 100% pasting process during collisions of particles submitted to a vertical field of gravity.

Tridimensional fractal aggregate with variable fractal dimension -increasing from front to back-.




Bidimensional Fluids [Fluides Bidimensionnels] :

A bidimensional periodical fluid with a central obstacle and with display of velocity histograms.	A bidimensional periodical fluid with strictly identical initial velocities and with a vertically shifted central obstacle and with display of velocity histograms.	Tridimensional display of the particle density of a bidimensional periodical fluid with strictly identical initial velocities and with a vertically shifted central obstacle.	A higly symmetrical system (a bidimensional periodical fluid with strictly identical initial velocities and with a vertically shifted central obstacle) can lost very fast the memory of its initial symmetry -with display of velocity histograms-.	The particle trajectories inside a bidimensional periodical fluid with a vertically shifted central obstacle and during a short time period.
The particle trajectories inside a bidimensional periodical fluid with a vertically shifted central obstacle and over a long time period.




Tridimensional Fluids [Fluides Tridimensionnels] :

A tridimensional periodical fluid with display of velocity histograms.




Thermodynamics [Thermodynamique] :

Particle collisions without energy loss in a bidimensional space with display of the velocity histogram.	Particle collisions without energy loss in a tridimensional space with display of the velocity histogram.	Particle collisions with energy loss in a tridimensional space with display of the velocity histogram.	Expansion of a gas inside a bidimensional rectangular box with display of velocity histograms.	Expansion of a gas inside a bidimensional circular box with display of velocity histograms.
Diffusion between two boxes (initial conditions : the left one is empty, the right one contains 256 particles), with collisions and display of the gravity center -white particle-.	Diffusion between two boxes (initial conditions : the left one is empty, the right one contains 256 particles), with collisions and a time-dependent geometry.	A leaking bidimensional piston with following initial conditions : high temperature particles at left and low temperature particles at right.	A perfect bidimensional piston with following initial conditions : still piston, high temperature particles at left and low temperature particles at right, and with display of velocity histograms.	A perfect bidimensional piston with following initial conditions : still piston, high temperature particles at left and low temperature particles at right, in a bidimensional box submitted to a vertical field of gravity.
A perfect bidimensional piston with following initial conditions : moving piston, same temperature for the particles at left and at right, and with display of velocity histograms.	A perfect tridimensional piston with following initial conditions : still piston, high temperature particles at left and low temperature particles at right, in a tridimensional box submitted to a vertical field of gravity.	A perfect tridimensional piston with following initial conditions : still piston, high temperature particles at left and low temperature particles at right, in a tridimensional box submitted to a vertical field of gravity.	A porous membrane with following initial conditions : high temperature particles at left and low temperature particles at right.	Particles in a bidimensional box submitted to a vertical field of gravity.
The trajectories of high temperature particles in a bidimensional box submitted to a vertical field of gravity.	Particles in a bidimensional box submitted to an attractive central field of gravity.	The trajectories of high temperature particles in a bidimensional box submitted to an attractive central field of gravity.	Particles in a bidimensional box submitted to a repulsive central field of gravity.	The trajectories of high temperature particles in a bidimensional box submitted to a repulsive central field of gravity.




Bidimensional Spin Networks [Réseaux Bidimensionnels de Spins] :

[more information about the Bidimensional Ising Model -in french-]



The bidimensional Ising Model with 2-state spins, temperature=0.2 and regular initial conditions (a chessboard).	The bidimensional Ising Model with 2-state spins, temperature=0.2, a non-isotropic kernel and regular initial conditions (a chessboard).	The bidimensional Ising Model with 2-state spins, a centred gaussian temperature field (minimum=0.035,maximum=0.6) and regular initial conditions (a chessboard).	The bidimensional Ising Model with 2-state spins, temperature=0.2 and random initial conditions.	The bidimensional Ising Model with 2-state spins, temperature=0.2, a non-isotropic kernel and random initial conditions.
The bidimensional Ising Model with 2-state spins, a centred gaussian temperature field (minimum=0.035,maximum=0.6) and random initial conditions.	The bidimensional Ising Model after 1000 iterations, with 2-state spins, random initial conditions and for different temperatures around the critical temperature.	The bidimensional Ising Model after 1000 iterations, with 2-state spins, random initial conditions and for a temperature equal to the critical temperature.	The scale invariance of the bidimensional Ising Model after 20000 iterations on a 5000x5000 lattice, with 2-state spins, random initial conditions and for a temperature equal to the critical temperature.




Tridimensional Spin Networks [Réseaux Tridimensionnels de Spins] :

[more information about the Tridimensional Ising Model -in french-]



The tridimensional Ising Model with 2-state spins, temperature=0.1 and random initial conditions.	The tridimensional Ising Model with 2-state spins, temperature=0.1 and random initial conditions.	The tridimensional Ising Model with 2-state spins, temperature=0.2 and random initial conditions.	The tridimensional Ising Model with 2-state spins, temperature=0.2 and random initial conditions.	The tridimensional Ising Model with 2-state spins, temperature=0.2, random initial conditions and an increasing number of iterations -from 100 to 4000-.
The tridimensional Ising Model with 2-state spins, temperature=0.4 and random initial conditions.	The tridimensional Ising Model with 2-state spins, temperature=0.4 and random initial conditions.

Copyright (c) Jean-François Colonna, 2001-2013.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2001-2013.