What is Canonical ensemble: Definition and 58 Discussions
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.
The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: T). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: N) and the system's volume (symbol: V), each of which influence the nature of the system's internal states. An ensemble with these three parameters is sometimes called the NVT ensemble.
The canonical ensemble assigns a probability P to each distinct microstate given by the following exponential:
P
=
e
(
F
−
E
)
/
(
k
T
)
,
{\displaystyle P=e^{(F-E)/(kT)},}
where E is the total energy of the microstate, and k is Boltzmann's constant.
The number F is the free energy (specifically, the Helmholtz free energy) and is a constant for the ensemble. However, the probabilities and F will vary if different N, V, T are selected. The free energy F serves two roles: first, it provides a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function F(N, V, T).
An alternative but equivalent formulation for the same concept writes the probability as
P
=
1
Z
e
−
E
/
(
k
T
)
,
{\displaystyle \textstyle P={\frac {1}{Z}}e^{-E/(kT)},}
using the canonical partition function
Z
=
e
−
F
/
(
k
T
)
{\displaystyle \textstyle Z=e^{-F/(kT)}}
rather than the free energy. The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations.
Historically, the canonical ensemble was first described by Boltzmann (who called it a holode) in 1884 in a relatively unknown paper. It was later reformulated and extensively investigated by Gibbs in 1902.
Homework Statement
To calculate internal energy for a system of non interacting S.H.O's in 1 dimension at constant T & u(chem potential) using grand partition function
The Attempt at a Solution
L(Grnd prtition fn)= Summation(z^N)*[Z(TVN)] where z = fugacity , Z = partition fn of...
Homework Statement
In the canonical ensemble, the probability that a system is in state r is given by
P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)}
where g_i is the multiplicity of state i. This is confusing me because I thought
P_i = \frac{g_i}{\sum_i g_i} = states...
Homework Statement
Grand Canonical ensemble problem- A surface of N sites that can have 0,1,2 atoms. It costs no energy to adsorb an atom. Grand canonical problem therefore in contact with particle reservoir. Assume \mu chem potentiol and temp T.
What is probability for site to be empty,1,or 2...
I've been studying and thinking about statistical physics for a couple days now... and what bothers me is the grand canonical partition function. Namely that for a system with fixed chemical potential and energy \epsilon_i the probability of having N_i particle in that state is proportional to...
Hey folks, I really need help setting up a canonical ensemble.
I am using the einstein model and have an energy:
E_\nu=A + \sum_{i=1}^{2N} \hbar\omega n_i+\sum_{j=1}^{N} \hbarh\omega n_j
where A is some constant.
Now, I need to build the partition function which I 'think' looks like...
Hi there,
I have a system with the following energy using the einstein model:
E_\nu=\sum_{i=1}^{2N} h\omega n_i+\sum_{j=1}^{N} h\omega n_j
I need to set up a canonical ensemble for this.
How would I write the partition function please?
I am taking a statictical mechanics course, and one thing bothers me. I am not sure when we should use the normal partition function (Z) and when the grand partition function (twisty Z). In particular, why can we not use grand partition function when we are considering the following system...