8. Energy and Entropy of Finite Quantum Systems

In this section, we discuss the results of Section 6 on the energy and entropy of quantized fields in finite volumes at non-zero temperature.

It was recently suggested [2] that the entropy of any finite quantum mechanical system should be bounded by

where is the energy of the system, and is the radius of a circumscribing sphere. The equality is realized for a black hole. Bound (8.1) would imply an absolute lower limit on the energy required to transmit information (negentropy) at a particular rate [48].

[ Figure 11 ] Energy and entropy of an electromagnetic field in a perfectly-conducting three-dimensional cubic cavity. The ratio exhibits a definite maximum.

Since the entropy of a system cannot be negative, the validity of the bound (8.1) requires that . However, Sections 3 and 4 show that a finite physical system can have . (16) The bound (8.1) is therefore incorrect. Figure 8.1 shows the energy and entropy of an electromagnetic field in a cubical three-dimensional cavity. Notice that the maximum value of is achieved at the point for which , as expected from Ref. [2]. Figure 8.2 shows and for a cavity with side lengths in the ratio 1: 1: 4, and provides an example of a system for which (8.1) fails.

[ Figure 12 ] Energy and entropy of an electromagnetic field in a perfectly conducting three-dimensional cavity with sides in the ratio 1: 1:4. The energy is negative for sufficiently low temperatures, and no bound on exists.

The existence of a bound such as (8.1) is suggested by the second law of black hole dynamics [2]. This law is however violated by the presence of negative energies through the failure of the dominant energy condition.

Other reasons for and examples of the violation of (8.1) have very recently been given in [49].

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