# How is energy related to entropy?

I have been reading the book "The Black Hole War," by L.Susskind, which states:

Temperature is the increase of a system's energy when an entropy bit is added.

Until now I was thinking that, on the contrary, it is the entropy that is increased when energy is added. No matter how you define entropy as hidden information or diffusion of energy or log of microstates or as heat divided by absolute temperature, the arrow of causality is from energy to entropy and not the other way around. I also don't understand why adding of entropy by itself increases energy.

What is the meaning of this definition?

• You could equally interpret it as how much energy is required to increase the entropy by 1 unit. – lemon Mar 9 '17 at 11:58

The sentence is really just translating the equation

$$dU = TdS$$

into words, although it is also neglecting the other components of energy changes, $-PdV + \mu dN$, which are the energy changes due to changes in volume and chemical composition.

If you rearrange the equation and hold volume, pressure and composition constant, you get:

$$\frac{dU}{dS}\biggr\rvert_{V,P,N} = T$$

And that is what the sentence is saying. Temperature is the change in energy due to the change in entropy. And since there is no negative sign, it is phrased as a positive -- energy increases when entropy is added.

Now, if temperature is constant, then what you said in your question is correct -- if you double the energy, the entropy will also double. But the reverse is true as well. For a fixed temperature, if you double the entropy, the energy doubles also.

• Thank you for your answer but I always meet the definition of energy as inverted T. How diffusion of energy increases energy? Increase of energy should make energy spread. – veronika Mar 9 '17 at 11:57
• @veronika I'm not sure what you mean by inverted T exactly. Every definition of energy has it directly proportional to T -- $dU = T dS$, or $e = c_v T$, or $E = k_b T$. Can you show me an equation that has it otherwise? I'm also not sure what you mean by the diffusion of energy increasing energy... that isn't the case either. Total energy is conserved. If you have, say, a Gaussian pulse of energy, it will diffuse outwards and the peak will lower and the pulse gets wider. But the area under the curve is always the same, there is no increase or decrease in the total amount of energy. – tpg2114 Mar 9 '17 at 12:01
• I am abstract, definition of temperature not energy has the T inverted, 1/T. Diffusion of energy is entropy. If you have more energy the Gaussian pulse you are writing can expand but the opposite as you also write is not correct. Energy is more fundamental than entropy. – veronika Mar 9 '17 at 12:52
• @tpg2114 In his basic TD book, Schroeder (Intro to Thermal Physics, P88) uses the recipricol of T because he says "it's more convenient in practice, rarely do you ever have a formula for energy in terms of entropy, volume and the number of particles " and using the "normal," above definition is mainly for numerical examples. He spends the rest of the book using $1/T$. It's' a nice book, easy to follow imo and a widely recommended book, but this is the third time it has been called out in the questions I have seen here. It may be the book the OP has, or a book following a similiar idea. – user146020 Mar 9 '17 at 13:42
• Even Atkins in his popular science book Laws of Thermodynamics, says that it was a mistake to call $T$ as temperature, and $1/T$ would have been a more appropriate choice, since then our inability to reach infinite temperature ($1/T$ definition) would make immediate sense. – Deep Mar 10 '17 at 5:14