# Reference paper to support information — energy relation $\left(kT \ln2 \rm\frac{J}{bit}\right)\;.$

In answer to Maxwell's Demon Constant (Information-Energy equivalence) there is stated that one bit of information allows to perform $kT \cdot \ln2$ Joules of work. Which paper supports the thesis? (there are many publications on Maxwell daemon, Szilard engine, Landauer's principle).

See e.g. page 3 of

http://arxiv.org/abs/0707.3400

It's nonsensical to attribute this simple particular insight to a "discoverer"; all these considerations should be associated with Ludwig Boltzmann who knew the answer even though the information in physics was considered continuous at that time.

One may easily derive the result. For example, put one molecule of an ideal gas in a vessel, learn in which half of the vessel the molecule is (one bit of information), and put a barrier in the middle.

You will be able to allow the molecule to do the work and expand from $V/2$ to the original volume $V$. The molecule will do the work $$W = \int_{V/2}^V p\,dV = \int_{V/2}^V \frac{kT}{V}dV = kT \ln \frac{V}{V/2} = kT\ln 2$$ where I used $pV = NkT$ for $N=1$ molecule of an ideal gas. More generally, you don't have to consider ideal gas. Just recall how work is related to the free energy, $E-TS$. To reduce the entropy of a subsystem by one bit, i.e. by $k\ln 2$ (look at Boltzmann's tomb formula to know why it's this value), we have to do change $E-TS$ by $kT\ln 2$.

• I like nice simple and well explained answers +1 – Pygmalion Apr 19 '12 at 6:44
• One note though, are you using one symbol $E$ for two different quantities? – Pygmalion Apr 19 '12 at 6:50
• @LubošMotl: Doesn't this mean that information is more a controll of energy (which can be prepared via manipulating T), not the energy? – Mooncer Apr 19 '12 at 6:56
• Dear Pygmalion, I changed one $E$ to $W$. Steffen, information surely "is not" energy. They're different quantities. But much like pretty much any pair of quantities in physics, they may have various relationships and there may exist important true sentences in which both of them appear. At a fixed temperature $T$, as you said, 1 bit allows to perform $kT\ln 2$ of work. But that doesn't mean that energy and information is the same thing, e.g. because $T$ isn't a universal constant. It's the temperature, another quantity. In a similar way, we don't say that voltage and current is the same thing – Luboš Motl Apr 19 '12 at 12:22
• Nice answer, really. Could you please explain further why you mention free energy $E-TS$ and the necessity to change it by $KTln2$ in order to reduce entropy by one bit? As I understand, Szilárd thought measurement process required dissipation, hence the paradox disolves. However, as shown by Bennett, acquisition of information does not necessarily involve dissipation. It's the necessity to return to a suitable state for measurement which constitutes an irreversible process (since it involves logical irreversibility). What do you think about this? – cacosomoza Jul 7 '13 at 12:55