Information, Entropy and Holographic Screen
Problem 1:
Show that information density on holographic screen is limited by the value $\sim 10^{69} \: \mbox{bit/m}^2.$
Problem 2:
Choosing the Hubble sphere as a holographic screen, find its area in the de Sitter model (recall that in this model the Universe dynamics is determined by the cosmological constant $\Lambda >0$).
$$ A=\frac{12\pi }{\Lambda }. $$
Problem 3:
Find the entropy of a quantum system composed of $N$ spin-$1/2$ particles.
Entropy of a quantum system equals to logarithm of the number of degrees of freedom $N,$ i.e. to logarithm of the dimension $\dim H$ of the Hilbert space $$ S=\ln N=\ln (\dim H). $$ For the system of $N$ spin-$1/2$ particles $\dim H = 2^N$ and $$ S=N\ln 2. $$
Problem 4:
Consider a 3D lattice of spin-$1/2$ particles and prove that entropy of such a system defined in a standard way is proportional to its volume.
Let the lattice occupy volume ${{L}^{3}}$, and its spatial period equals to $l$. Entropy of the system is proportional to the number of (quantum) states $N={{2}^{n}},\ n={{\left( \frac{L}{l} \right)}^{3}}$. Therefore \[S\propto \ln N\propto {{\left( \frac{L}{l} \right)}^{3}}\propto V.\]
Problem 5:
Find the change of the holographic screen area when it is crossed by a spin-$1/2$ particle.
Let all the punctures of the holographic screen be due its crossing by spin-$1/2$ particles. Then for $N$ punctures $$ N=\frac{A{{c}^{3}}}{4G\hbar \ln 2}. $$ The crossing leads to an additional puncture and therefore to change of the surface area equal to $$ \Delta A=\frac{\hbar G}{{{c}^{3}}}4\ln 2. $$
Problem 6:
What entropy change corresponds to the process described in the previous problem?
$\Delta S=\ln 2$ (see problem).
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