Interpretation of the Boltzmann factor and partition function
$$p_i = frac{ expleft(-frac{epsilon _i}{k_BT} right)}{Z} $$
$$ Z= sum_{i} expleft(-frac{epsilon _i}{k_BT} right)$$
A) Is $p_i$ the probability of the system having an energy equal to $epsilon_i$? (Probability to be in any of the many microstates that have energy $epsilon_i$).
B) Or is $p_i$ the probability of the system being in one particular microstate which happens to have energy $epsilon_i$? (This microstate is not the only microstate with the same energy).
If A) is correct then:
$$ Z= sum_{epsilon_i} expleft(-frac{epsilon _i}{k_BT} right)$$
If B) is correct then:
$$ Z= sum_{epsilon_i} Omega_iexpleft(-frac{epsilon _i}{k_BT} right),$$
where $Omega_i$ is the multiplicity of the macrostate of energy $epsilon_i$.
From the derivation of the Boltzmann distribution I am inclined to understand it as B). But I have never seen the multiplicity in the partition function.
What is the correct interpretation of the Boltzmann distribution?
statistical-mechanics probability partition-function
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$$p_i = frac{ expleft(-frac{epsilon _i}{k_BT} right)}{Z} $$
$$ Z= sum_{i} expleft(-frac{epsilon _i}{k_BT} right)$$
A) Is $p_i$ the probability of the system having an energy equal to $epsilon_i$? (Probability to be in any of the many microstates that have energy $epsilon_i$).
B) Or is $p_i$ the probability of the system being in one particular microstate which happens to have energy $epsilon_i$? (This microstate is not the only microstate with the same energy).
If A) is correct then:
$$ Z= sum_{epsilon_i} expleft(-frac{epsilon _i}{k_BT} right)$$
If B) is correct then:
$$ Z= sum_{epsilon_i} Omega_iexpleft(-frac{epsilon _i}{k_BT} right),$$
where $Omega_i$ is the multiplicity of the macrostate of energy $epsilon_i$.
From the derivation of the Boltzmann distribution I am inclined to understand it as B). But I have never seen the multiplicity in the partition function.
What is the correct interpretation of the Boltzmann distribution?
statistical-mechanics probability partition-function
add a comment |
$$p_i = frac{ expleft(-frac{epsilon _i}{k_BT} right)}{Z} $$
$$ Z= sum_{i} expleft(-frac{epsilon _i}{k_BT} right)$$
A) Is $p_i$ the probability of the system having an energy equal to $epsilon_i$? (Probability to be in any of the many microstates that have energy $epsilon_i$).
B) Or is $p_i$ the probability of the system being in one particular microstate which happens to have energy $epsilon_i$? (This microstate is not the only microstate with the same energy).
If A) is correct then:
$$ Z= sum_{epsilon_i} expleft(-frac{epsilon _i}{k_BT} right)$$
If B) is correct then:
$$ Z= sum_{epsilon_i} Omega_iexpleft(-frac{epsilon _i}{k_BT} right),$$
where $Omega_i$ is the multiplicity of the macrostate of energy $epsilon_i$.
From the derivation of the Boltzmann distribution I am inclined to understand it as B). But I have never seen the multiplicity in the partition function.
What is the correct interpretation of the Boltzmann distribution?
statistical-mechanics probability partition-function
$$p_i = frac{ expleft(-frac{epsilon _i}{k_BT} right)}{Z} $$
$$ Z= sum_{i} expleft(-frac{epsilon _i}{k_BT} right)$$
A) Is $p_i$ the probability of the system having an energy equal to $epsilon_i$? (Probability to be in any of the many microstates that have energy $epsilon_i$).
B) Or is $p_i$ the probability of the system being in one particular microstate which happens to have energy $epsilon_i$? (This microstate is not the only microstate with the same energy).
If A) is correct then:
$$ Z= sum_{epsilon_i} expleft(-frac{epsilon _i}{k_BT} right)$$
If B) is correct then:
$$ Z= sum_{epsilon_i} Omega_iexpleft(-frac{epsilon _i}{k_BT} right),$$
where $Omega_i$ is the multiplicity of the macrostate of energy $epsilon_i$.
From the derivation of the Boltzmann distribution I am inclined to understand it as B). But I have never seen the multiplicity in the partition function.
What is the correct interpretation of the Boltzmann distribution?
statistical-mechanics probability partition-function
statistical-mechanics probability partition-function
edited yesterday
Qmechanic♦
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Daniel DuqueDaniel Duque
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To the first question, the answer is B: $p_i$ is the probability of being in the $i$-th microstate, which happens to have an energy $varepsilon_i$. However, microstates other than the $i$-th one may also have an energy $varepsilon_i$.
The reason you never see the multiplicity in the partition function is because you are probably looking at summations done over the microstates: $$Z=sum_i e^{-frac{varepsilon_i}{k_BT}}$$ instead of over the internal energies as you’ve written above.
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
add a comment |
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1 Answer
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oldest
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To the first question, the answer is B: $p_i$ is the probability of being in the $i$-th microstate, which happens to have an energy $varepsilon_i$. However, microstates other than the $i$-th one may also have an energy $varepsilon_i$.
The reason you never see the multiplicity in the partition function is because you are probably looking at summations done over the microstates: $$Z=sum_i e^{-frac{varepsilon_i}{k_BT}}$$ instead of over the internal energies as you’ve written above.
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
add a comment |
To the first question, the answer is B: $p_i$ is the probability of being in the $i$-th microstate, which happens to have an energy $varepsilon_i$. However, microstates other than the $i$-th one may also have an energy $varepsilon_i$.
The reason you never see the multiplicity in the partition function is because you are probably looking at summations done over the microstates: $$Z=sum_i e^{-frac{varepsilon_i}{k_BT}}$$ instead of over the internal energies as you’ve written above.
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
add a comment |
To the first question, the answer is B: $p_i$ is the probability of being in the $i$-th microstate, which happens to have an energy $varepsilon_i$. However, microstates other than the $i$-th one may also have an energy $varepsilon_i$.
The reason you never see the multiplicity in the partition function is because you are probably looking at summations done over the microstates: $$Z=sum_i e^{-frac{varepsilon_i}{k_BT}}$$ instead of over the internal energies as you’ve written above.
To the first question, the answer is B: $p_i$ is the probability of being in the $i$-th microstate, which happens to have an energy $varepsilon_i$. However, microstates other than the $i$-th one may also have an energy $varepsilon_i$.
The reason you never see the multiplicity in the partition function is because you are probably looking at summations done over the microstates: $$Z=sum_i e^{-frac{varepsilon_i}{k_BT}}$$ instead of over the internal energies as you’ve written above.
edited yesterday
answered yesterday
Riley Scott JacobRiley Scott Jacob
2937
2937
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
add a comment |
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I would not say that it is bizarre to write the partition function a a sum over energies. I have certainly seen it done. It is common, for example to do this as a precursor to moving representation of the partition function as an integral over energy levels, in which the multiplicity $g_i$ becomes a contribution to the density of states $g(epsilon)$ states.
– By Symmetry
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
I do apologize - I am only an undergraduate and have never seen it as a sum over the energies in my limited experience. I will amend my statement.
– Riley Scott Jacob
yesterday
add a comment |
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