## Study of phase transitions

Working from Remoton.

I find the cost of 1.2*10^8 random configurations.

>> generacostes_rand(‘Costes_rand’,12,4)
>> [histog,bins]=costes_rand2histog(‘Costes_rand’,180:.5:260);
>> plot(bins,histog)

>> pos2coste(todas.A/29.3,todas.M/29.3+todas.S,todas.f,todas.pos_real,.5)
ans =
119.6195

Actual configuration has a cost much lower than those we get randomly. Therefore, this is not a valid scheme to see phase transitions in a range of temperatures similar to that of the real worm. Nevertheless, I will study the phase transitions with these data:

>> [costemedio,varianza]=costes_rand2mediavarianza(‘Costes_rand’,.1:.1:5,1);
>> T=.1:.1:5;
>> plot(T,costemedio)

>> plot(T,varianza)

There is this strange maximum. A possible explanation is that the effect of the exponential is to remove the peak of the cost distribution, producing something more uniform, which has larger variance. No me lo creo ni yo, pero bueno.

Specific heat:
>> plot(T,varianza./T.^2)

Uooooohhh!!!

I do not quite understand this. The cost histogram looks too boring to be hiding a phase transition. I run a control using normally distributed costs:

>> clear
>> costes=randn(1,10^7)*10;
>> save prueba1 costes
>> [costemedio,varianza]=costes_rand2mediavarianza(‘prueba’,.1:.1:5,1);
>> T=.1:.1:5;
>> plot(T,costemedio)

>> plot(T,varianza)

>> plot(T,varianza./T.^2)

Veeeery similar. So I do not think there is anything especial in the C. elegans cost.

Why is this apparent phase transition taking place? An older post showed that local minima were needed to get a phase transition. But anyway, the cost histogram for a single well is very different to the one we get here:

>> x=-1:.01:1;
coste=(abs(x)).^2;
>> hist(coste)

Indeed, the effect of a local minimum is to create a peak in the histogram of costs:

>> x=-1:.00001:1;
coste=(abs(x-.5)).^.5;
subplot(1,2,1)
plot(x,coste)
subplot(1,2,2)
hist(coste)

>> x=-1:.00001:1;
coste=(abs(x-.5)).^.5+.7*(abs(x+.6)).^.5;
subplot(1,2,1)
plot(x,coste)
subplot(1,2,2)
hist(coste)

## New figures for E. coli (definitive)

For maintenance, the final figure is the one in previous post.

Summary of results:

 D=0.05 D=0.1 D=0.2 D=0.3 D=0.4 Delta for main plots 1.5 1.99 2.54 4.22 5.31 p_growth (main plots) 0.01 0.0058 2.0000e-004 4.8000e-004 6.2000e-004 p_ATP (main plots) NaN 0.1739 0.2399 0.0854 0.1211 max(p_growth) (all Deltas) 0.0187 0.0383 0.0517 0.0024 0.0011 min(p_ATP) (all Deltas) NaN 0.0920 0.0302 0.0280 0.0245

Growth 0.05 (maybe not final, maybe will not be included):

>> fig_pap_coli_13(1,0,.05)

Growth 0.1:
>> fig_pap_coli_13(1,0,.1)

>> fig_pap_coli_13(1,0,.2)

>> fig_pap_coli_13(1,0,.3)

>> fig_pap_coli_13(1,0,.4)