The third and higher order entropies also exhibit unphysical phase transitions. The derivative of the third order entropy has transitions at q=1/n, 2/N, 1-1/N, and 1-2/N. Note again that it is difficult to interpret such
transitions physically because the systems described away from the discrete values of 
must have fractional numbers of bits to be realized. In general, the derivative of the mth order entropy exhibits a phase transition at 2(m-1) locations, up to 
.
Graphs of the corrections 
, 
as a function of N at q=1/N appear in figure 4.1, and as a function of bit density q for N=10 in figure 4.2. Note that complex values occur at unphysical values of q only when too few or too many one bits are sought in the observed region, or for N less than the number of bits needed to make the entropies exist.
The variation of the entropy differences at physical values of q are of interest. Graphs of these variations as a function of the order of the entropy and N=10 appear next. The differences are normalized by the order of the
entropy. In figure 4.5 the entropy differences between the points q=k/N and q=(k-1)/N are plotted for 
. These figures demonstrate that even for this simple system it is important to consider the
higher order functions of the probability distributions of observed states in certain regions of the parameter space.
Figure 4.1: Correction terms , m=2,3,4 at q=1/N as a function of the length of the strings, N. Wider spaces in the dashing indicates lower m. Plot indicates that thermodynamic limit of corrections is zero. Low N behavior is complex due to having strings shorter than needed to define the entropy involved.
Figure 4.2: Correction terms , m=2,3,4 at N=10 as a function of q. Wider spaces in the dashing indicates lower m. Off q=k/N for integer k behavior is complex due to having nonphysical numbers of bits, fractional numbers of bits.
Figure 4.3: Second order correction term 
at N=10 as a function of q. Real part is solid. Imaginary part is dashed. Line crosses curve at divergent derivative.
Figure 4.4: Derivative of second order correction term at N=10 as a function of q near 1/N. Real part is solid. Imaginary part is dashed.
Figure 4.5: Entropy differences. Entropy order on x axis.
The quantity plotted is
for 
and
strings of length N=10.
Higher graph indicates lower B.
These are effectively the derivative functions at small
one bit counts per string.