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Made a mistake in writing the ensemble variables for the canonical case
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michael b
michael b
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I'm reading through Pathria and in chapter 3 he presents a derivation of the canonical ensemble via combinatorics. I'll summarize the key points of his derivation here, and I'll state my questions at the end.

Pathria's Derivation + some in between steps

We have $\mathcal{N}$ systems with macro state ($T,V,E$$N,V,T$) distributed among all available energy levels, with $n_r$ being the number of systems with energy $E_r$ such that, $$\sum_{r}n_r=\mathcal{N} \tag{1}$$ $$\sum_{r}E_rn_r=\mathcal{E} \tag{2}$$ There are $W$ ways of distributing those $\mathcal{N}$ systems among the energy levels, $$W=\frac{\mathcal{N}!}{n_0!n_1!...} \tag{3}$$ Using some rules of logarithms and Stirling's approximation we get, $$\ln W=\mathcal{N}\ln \mathcal{N}-\sum_r n_r \ln n_r \tag{4}$$ If we disturb the distribution set such that the set is $\{n_r+\delta n_r\}$ and calculate the disturbance of $\ln W$, we get, $$\delta \ln W = -\sum_r (1 + \ln n_r)\delta n_r = f(\delta n_r) \tag{5}$$ Based on the conditions in (1) and (2), the total number of systems $\mathcal{N}$ is unaltered and the total energy of all systems is still $\mathcal{E}$ so that, $$\sum_r \delta n_r =0 = g(\delta n_r) \tag{6}$$ $$\sum_r E_r \delta n_r =0 =h(\delta n_r)\tag{7}$$ Since the most probable distribution will occur when $\ln W$ is maximized, the disturbance $\delta \ln W$ should be minimized. This can be achieved by using Lagrange multipliers and considering the constraint equations (6) and (7) having the multipliers $\alpha$ and $\beta$ respectively. $$\mathcal{L}(\delta n_r, \alpha, \beta) = f(\delta n_r) - \alpha g(\delta n_r)-\beta h(\delta n_r) $$ $$=-\sum_r \delta n_r(1 + \ln n_r + \alpha + \beta E_r) \tag{8}$$ $$\frac{\partial \mathcal{L}}{\partial \delta n_r} =-\sum_r (1 + \ln n_r + \alpha + \beta E_r) = 0 \tag{9}$$ This ultimately gives us, $$n_r=Ce^{-\beta E_r}$$ And, $$\frac{n_r}{\mathcal{N}}=\frac{Ce^{-\beta E_r}}{\sum_r Ce^{-\beta E_r}}=\frac{e^{-\beta E_r}}{\sum_r e^{-\beta E_r}}=P(E_r) \tag{10}$$ Okay, great, so we have the canonical ensemble from a combinatorics vantage point.

Questions:

  1. Is the canonical ensemble "approximate"? We only get (4) due to Stirling's approximation. When we derive the canonical ensemble using $d\rho/dt = 0$ and $\{\rho, H\}=0$, we get (10) precisely. Pathria does another derivation involving a Taylor series expansion of $\ln \Omega_{heat bath}$ near $E_{heatbath}=E_{system}$ and we only take the first two terms. Does this imply there are "more terms" in the canonical ensemble?
  2. To arrive at (5) I basically took the derivative of (4) with respect to $n_r$ and then multiplied by our variation $\delta n_r$. Is this legitimate? Is there a name on this type of analysis?
  3. Since $\delta n_r$ is discrete, I would assume it can't be zero, and in principle should be $\delta n_r = 1$. In what sense are (6) and (7) true then? Can we make the claim that $\mathcal{N}$ and $\mathcal{E}$ are unaltered?
  4. Something feels "wrong" about constructing the Lagrangian (8) of a function which is dependent on a variation. In (9) we take the partial derivative of (8) with respect to the variation, which doesn't seem sound mathematically. Can we treat the variation as a truly independent variable here?

Thanks for reading. On a side note, Pathria has been incredible to read, and has tied together many theoretical pieces of stat. mech. for me.

I'm reading through Pathria and in chapter 3 he presents a derivation of the canonical ensemble via combinatorics. I'll summarize the key points of his derivation here, and I'll state my questions at the end.

Pathria's Derivation + some in between steps

We have $\mathcal{N}$ systems with macro state ($T,V,E$) distributed among all available energy levels, with $n_r$ being the number of systems with energy $E_r$ such that, $$\sum_{r}n_r=\mathcal{N} \tag{1}$$ $$\sum_{r}E_rn_r=\mathcal{E} \tag{2}$$ There are $W$ ways of distributing those $\mathcal{N}$ systems among the energy levels, $$W=\frac{\mathcal{N}!}{n_0!n_1!...} \tag{3}$$ Using some rules of logarithms and Stirling's approximation we get, $$\ln W=\mathcal{N}\ln \mathcal{N}-\sum_r n_r \ln n_r \tag{4}$$ If we disturb the distribution set such that the set is $\{n_r+\delta n_r\}$ and calculate the disturbance of $\ln W$, we get, $$\delta \ln W = -\sum_r (1 + \ln n_r)\delta n_r = f(\delta n_r) \tag{5}$$ Based on the conditions in (1) and (2), the total number of systems $\mathcal{N}$ is unaltered and the total energy of all systems is still $\mathcal{E}$ so that, $$\sum_r \delta n_r =0 = g(\delta n_r) \tag{6}$$ $$\sum_r E_r \delta n_r =0 =h(\delta n_r)\tag{7}$$ Since the most probable distribution will occur when $\ln W$ is maximized, the disturbance $\delta \ln W$ should be minimized. This can be achieved by using Lagrange multipliers and considering the constraint equations (6) and (7) having the multipliers $\alpha$ and $\beta$ respectively. $$\mathcal{L}(\delta n_r, \alpha, \beta) = f(\delta n_r) - \alpha g(\delta n_r)-\beta h(\delta n_r) $$ $$=-\sum_r \delta n_r(1 + \ln n_r + \alpha + \beta E_r) \tag{8}$$ $$\frac{\partial \mathcal{L}}{\partial \delta n_r} =-\sum_r (1 + \ln n_r + \alpha + \beta E_r) = 0 \tag{9}$$ This ultimately gives us, $$n_r=Ce^{-\beta E_r}$$ And, $$\frac{n_r}{\mathcal{N}}=\frac{Ce^{-\beta E_r}}{\sum_r Ce^{-\beta E_r}}=\frac{e^{-\beta E_r}}{\sum_r e^{-\beta E_r}}=P(E_r) \tag{10}$$ Okay, great, so we have the canonical ensemble from a combinatorics vantage point.

Questions:

  1. Is the canonical ensemble "approximate"? We only get (4) due to Stirling's approximation. When we derive the canonical ensemble using $d\rho/dt = 0$ and $\{\rho, H\}=0$, we get (10) precisely. Pathria does another derivation involving a Taylor series expansion of $\ln \Omega_{heat bath}$ near $E_{heatbath}=E_{system}$ and we only take the first two terms. Does this imply there are "more terms" in the canonical ensemble?
  2. To arrive at (5) I basically took the derivative of (4) with respect to $n_r$ and then multiplied by our variation $\delta n_r$. Is this legitimate? Is there a name on this type of analysis?
  3. Since $\delta n_r$ is discrete, I would assume it can't be zero, and in principle should be $\delta n_r = 1$. In what sense are (6) and (7) true then? Can we make the claim that $\mathcal{N}$ and $\mathcal{E}$ are unaltered?
  4. Something feels "wrong" about constructing the Lagrangian (8) of a function which is dependent on a variation. In (9) we take the partial derivative of (8) with respect to the variation, which doesn't seem sound mathematically. Can we treat the variation as a truly independent variable here?

Thanks for reading. On a side note, Pathria has been incredible to read, and has tied together many theoretical pieces of stat. mech. for me.

I'm reading through Pathria and in chapter 3 he presents a derivation of the canonical ensemble via combinatorics. I'll summarize the key points of his derivation here, and I'll state my questions at the end.

Pathria's Derivation + some in between steps

We have $\mathcal{N}$ systems with macro state ($N,V,T$) distributed among all available energy levels, with $n_r$ being the number of systems with energy $E_r$ such that, $$\sum_{r}n_r=\mathcal{N} \tag{1}$$ $$\sum_{r}E_rn_r=\mathcal{E} \tag{2}$$ There are $W$ ways of distributing those $\mathcal{N}$ systems among the energy levels, $$W=\frac{\mathcal{N}!}{n_0!n_1!...} \tag{3}$$ Using some rules of logarithms and Stirling's approximation we get, $$\ln W=\mathcal{N}\ln \mathcal{N}-\sum_r n_r \ln n_r \tag{4}$$ If we disturb the distribution set such that the set is $\{n_r+\delta n_r\}$ and calculate the disturbance of $\ln W$, we get, $$\delta \ln W = -\sum_r (1 + \ln n_r)\delta n_r = f(\delta n_r) \tag{5}$$ Based on the conditions in (1) and (2), the total number of systems $\mathcal{N}$ is unaltered and the total energy of all systems is still $\mathcal{E}$ so that, $$\sum_r \delta n_r =0 = g(\delta n_r) \tag{6}$$ $$\sum_r E_r \delta n_r =0 =h(\delta n_r)\tag{7}$$ Since the most probable distribution will occur when $\ln W$ is maximized, the disturbance $\delta \ln W$ should be minimized. This can be achieved by using Lagrange multipliers and considering the constraint equations (6) and (7) having the multipliers $\alpha$ and $\beta$ respectively. $$\mathcal{L}(\delta n_r, \alpha, \beta) = f(\delta n_r) - \alpha g(\delta n_r)-\beta h(\delta n_r) $$ $$=-\sum_r \delta n_r(1 + \ln n_r + \alpha + \beta E_r) \tag{8}$$ $$\frac{\partial \mathcal{L}}{\partial \delta n_r} =-\sum_r (1 + \ln n_r + \alpha + \beta E_r) = 0 \tag{9}$$ This ultimately gives us, $$n_r=Ce^{-\beta E_r}$$ And, $$\frac{n_r}{\mathcal{N}}=\frac{Ce^{-\beta E_r}}{\sum_r Ce^{-\beta E_r}}=\frac{e^{-\beta E_r}}{\sum_r e^{-\beta E_r}}=P(E_r) \tag{10}$$ Okay, great, so we have the canonical ensemble from a combinatorics vantage point.

Questions:

  1. Is the canonical ensemble "approximate"? We only get (4) due to Stirling's approximation. When we derive the canonical ensemble using $d\rho/dt = 0$ and $\{\rho, H\}=0$, we get (10) precisely. Pathria does another derivation involving a Taylor series expansion of $\ln \Omega_{heat bath}$ near $E_{heatbath}=E_{system}$ and we only take the first two terms. Does this imply there are "more terms" in the canonical ensemble?
  2. To arrive at (5) I basically took the derivative of (4) with respect to $n_r$ and then multiplied by our variation $\delta n_r$. Is this legitimate? Is there a name on this type of analysis?
  3. Since $\delta n_r$ is discrete, I would assume it can't be zero, and in principle should be $\delta n_r = 1$. In what sense are (6) and (7) true then? Can we make the claim that $\mathcal{N}$ and $\mathcal{E}$ are unaltered?
  4. Something feels "wrong" about constructing the Lagrangian (8) of a function which is dependent on a variation. In (9) we take the partial derivative of (8) with respect to the variation, which doesn't seem sound mathematically. Can we treat the variation as a truly independent variable here?

Thanks for reading. On a side note, Pathria has been incredible to read, and has tied together many theoretical pieces of stat. mech. for me.

Source Link
michael b
michael b
  • 822
  • 5
  • 14

Canonical Ensemble and Combinatorics

I'm reading through Pathria and in chapter 3 he presents a derivation of the canonical ensemble via combinatorics. I'll summarize the key points of his derivation here, and I'll state my questions at the end.

Pathria's Derivation + some in between steps

We have $\mathcal{N}$ systems with macro state ($T,V,E$) distributed among all available energy levels, with $n_r$ being the number of systems with energy $E_r$ such that, $$\sum_{r}n_r=\mathcal{N} \tag{1}$$ $$\sum_{r}E_rn_r=\mathcal{E} \tag{2}$$ There are $W$ ways of distributing those $\mathcal{N}$ systems among the energy levels, $$W=\frac{\mathcal{N}!}{n_0!n_1!...} \tag{3}$$ Using some rules of logarithms and Stirling's approximation we get, $$\ln W=\mathcal{N}\ln \mathcal{N}-\sum_r n_r \ln n_r \tag{4}$$ If we disturb the distribution set such that the set is $\{n_r+\delta n_r\}$ and calculate the disturbance of $\ln W$, we get, $$\delta \ln W = -\sum_r (1 + \ln n_r)\delta n_r = f(\delta n_r) \tag{5}$$ Based on the conditions in (1) and (2), the total number of systems $\mathcal{N}$ is unaltered and the total energy of all systems is still $\mathcal{E}$ so that, $$\sum_r \delta n_r =0 = g(\delta n_r) \tag{6}$$ $$\sum_r E_r \delta n_r =0 =h(\delta n_r)\tag{7}$$ Since the most probable distribution will occur when $\ln W$ is maximized, the disturbance $\delta \ln W$ should be minimized. This can be achieved by using Lagrange multipliers and considering the constraint equations (6) and (7) having the multipliers $\alpha$ and $\beta$ respectively. $$\mathcal{L}(\delta n_r, \alpha, \beta) = f(\delta n_r) - \alpha g(\delta n_r)-\beta h(\delta n_r) $$ $$=-\sum_r \delta n_r(1 + \ln n_r + \alpha + \beta E_r) \tag{8}$$ $$\frac{\partial \mathcal{L}}{\partial \delta n_r} =-\sum_r (1 + \ln n_r + \alpha + \beta E_r) = 0 \tag{9}$$ This ultimately gives us, $$n_r=Ce^{-\beta E_r}$$ And, $$\frac{n_r}{\mathcal{N}}=\frac{Ce^{-\beta E_r}}{\sum_r Ce^{-\beta E_r}}=\frac{e^{-\beta E_r}}{\sum_r e^{-\beta E_r}}=P(E_r) \tag{10}$$ Okay, great, so we have the canonical ensemble from a combinatorics vantage point.

Questions:

  1. Is the canonical ensemble "approximate"? We only get (4) due to Stirling's approximation. When we derive the canonical ensemble using $d\rho/dt = 0$ and $\{\rho, H\}=0$, we get (10) precisely. Pathria does another derivation involving a Taylor series expansion of $\ln \Omega_{heat bath}$ near $E_{heatbath}=E_{system}$ and we only take the first two terms. Does this imply there are "more terms" in the canonical ensemble?
  2. To arrive at (5) I basically took the derivative of (4) with respect to $n_r$ and then multiplied by our variation $\delta n_r$. Is this legitimate? Is there a name on this type of analysis?
  3. Since $\delta n_r$ is discrete, I would assume it can't be zero, and in principle should be $\delta n_r = 1$. In what sense are (6) and (7) true then? Can we make the claim that $\mathcal{N}$ and $\mathcal{E}$ are unaltered?
  4. Something feels "wrong" about constructing the Lagrangian (8) of a function which is dependent on a variation. In (9) we take the partial derivative of (8) with respect to the variation, which doesn't seem sound mathematically. Can we treat the variation as a truly independent variable here?

Thanks for reading. On a side note, Pathria has been incredible to read, and has tied together many theoretical pieces of stat. mech. for me.