Hello all, Why does the partition function of a free-particles system under the microcanonical ensemble approximate to V^N? V for volume; N for the number of particles; O for the partition function; int for integrate (d^N)r short for dr_1dr_2dr_3 ... dr_N (d^N)p short for dp_1dp_2dp_3 ... dp_N del for delta function H for Hamiltonion ( H=sum{p_i*p_i/(2m_i)} E for energy we do the integration in the phase space, so O(E) = int[(d^N)r]int[del{H-E}(d^N)p] I don''t understand why O(E) approximates to V^N. Can anyone explain it for me? Thanks in advance. Rex

The maths makes sense but I don''t recognise all of the terms - e.g. what is a ''partition function'', or a ''microcanonical ensemble'' ?

Perhaps describe the game problem you are trying to solve with this; it looks more complex than you need for simulating anything in a game, and in particular there are a few different fast and scalable solutions for simulating particle systems in games.

Yes, please explain the reason you''re asking this particular question in a game development forum. And show the work that you''ve done to try and derive the answer for yourself.

Particle systems are common for games and game development, of course, but your question doesn''t sound like a game development question. The question is a bit too theoretical. The specialized terminology (e.g., "microcanonical ensemble") makes me think of a University-level homework problem....read the forum FAQ.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.