Ensemble mean

The ensemble mean (also ensemble mean or cluster mean ) is a mean value from statistical physics . It can be used to calculate the mean value of a measured variable of all elements of an ensemble at a specific point in time. ${\displaystyle \langle \dots \rangle }$

use

For an ergodic system , the ensemble mean in a given ensemble is equal to the time mean determined over an infinitely long time . The ergodic hypothesis states that thermodynamic systems are ergodic and that the aforementioned equality applies to them.

definition

The ensemble mean of a size is given by: ${\displaystyle \langle A\rangle }$ ${\displaystyle A}$

${\displaystyle \langle A\rangle =\sum _{i\in I}p_{i}\cdot A_{i}=\sum _{i\in I}{\frac {e^{-{\frac {H_{i}}{k_{\mathrm {B} }\cdot T}}}\cdot A_{i}}{\sum _{i\in I}e^{-{\frac {H_{i}}{k_{\mathrm {B} }\cdot T}}}}}}$

With

• the probability in the canonical ensemble for a system with discrete states to find the system in the state : ${\displaystyle p}$ ${\displaystyle i}$

${\displaystyle p_{i}={\frac {e^{-{\frac {H_{i}}{k_{\mathrm {B} }\cdot T}}}}{Z}}}$,

• the canonical partition function :

${\displaystyle Z=\sum _{i\in I}e^{-{\frac {H_{i}}{k_{\mathrm {B} }\cdot T}}}}$

• the Boltzmann factor ${\displaystyle e^{-{\frac {H_{i}}{k_{\mathrm {B} }\cdot T}}}}$
• the Hamiltonian of the state ${\displaystyle H_{i}}$ ${\displaystyle i}$
• the Boltzmann constant ${\displaystyle k_{\mathrm {B} }}$
• the absolute temperature ${\displaystyle T}$
• the set of all micro-states . ${\displaystyle I}$

If the set of states can no longer be counted , but is continuous - for example, if the Hamiltonian of the system depends on continuous locations and continuous velocities, one goes from the sum to the integral by using the above discrete notation appropriately with the phase space element expanded, whereupon a Riemann integral is identified: ${\displaystyle I}$

${\displaystyle \langle A\rangle ={\frac {\int _{\mathbb {R} ^{6N}}{\frac {d^{3N}rd^{3N}p}{h^{3N}N!}}A\cdot e^{-{\frac {H({\vec {r}}_{1},\dots {\vec {r}}_{N},{\vec {p}}_{1},\dots {\vec {p}}_{N})}{k_{\mathrm {B} }\cdot T}}}}{\int _{\mathbb {R} ^{6N}}{\frac {d^{3N}rd^{3N}p}{h^{3N}N!}}\cdot e^{-{\frac {H({\vec {r}}_{1},\dots {\vec {r}}_{N},{\vec {p}}_{1},\dots {\vec {p}}_{N})}{k_{\mathrm {B} }\cdot T}}}}}}$

Further information

• Torsten Fließbach: Statistical Physics . 4th edition. Spektrum, Munich 2007, ISBN 978-3-8274-1684-1 .