content_image/343940.jpg
<img src='content_image/343940.jpg'>

of Fig. 3 can be fit with just one parameter namely µ ( N ) c . 

We have repeated the above analysis at larger values of L . We find the physics remains qualita- tively similar to the L = 2 case. In particular the average particle number jumps by one at critical values of µ . In Fig. 4 we show the average particle number as a function of µ at different values of L at β = 0 . 43 and β = 0 . 50 . In Fig.5 we plot the average particle number at L = 8 for all three values of β at different values of L$_{t}$ close to µ ( N ) c for N = 1 , 2 , 3 . The effective quantum mechanics description continues to fit all our data well as lo ng as µ is close to the critical values 

