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Steven Landsburg
Steven Landsburg
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The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

$$ C_n \, log(T/G) $$$$ Cn \, log(T/G) $$

where $G$ is the geometric mean. It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

$$ C_n \, log(T/G) $$

where $G$ is the geometric mean. It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

$$ Cn \, log(T/G) $$

where $G$ is the geometric mean. It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

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Tadashi
Tadashi
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The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity C$C$ and temperatures T_1,...T_n$T_1,\ldots,T_n$ and allow them to reach a final temperature T$T$. The first law of thermodynamics tells you that T$T$ is the arithmetic mean of the T_i$T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

Cn Log(T/G)$$ C_n \, log(T/G) $$

where G$G$ is the geometric mean. It follows that T > G$T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity C and temperatures T_1,...T_n and allow them to reach a final temperature T. The first law of thermodynamics tells you that T is the arithmetic mean of the T_i. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

Cn Log(T/G)

where G is the geometric mean. It follows that T > G.

I believe this argument was first made by P.T. Landsberg (no relation!).

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity $C$ and temperatures $T_1,\ldots,T_n$ and allow them to reach a final temperature $T$. The first law of thermodynamics tells you that $T$ is the arithmetic mean of the $T_i$. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

$$ C_n \, log(T/G) $$

where $G$ is the geometric mean. It follows that $T > G$.

I believe this argument was first made by P.T. Landsberg (no relation!).

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Steven Landsburg
Steven Landsburg
  • 23k
  • 5
  • 95
  • 153

The first and second laws of thermodynamics allow you to recover the inequality between the arithmetic and the geometric means: Bring together n identical heat reservoirs with heat capacity C and temperatures T_1,...T_n and allow them to reach a final temperature T. The first law of thermodynamics tells you that T is the arithmetic mean of the T_i. The second law of thermodynamics demands the non-negativity of the change in entropy, which is

Cn Log(T/G)

where G is the geometric mean. It follows that T > G.

I believe this argument was first made by P.T. Landsberg (no relation!).