Supplementary information 4:
The relation between the Shannon entropy S’ and the particle diameter Dp can be expressed as:

S '  k  f n ( Dp ) ln[ f n ( Dp )]dDp
0
(1)
where fn(Dp) is the number-based particle size distribution. Here, it is assumed that the entropy
function S’ is maximized in a dynamic equilibrium of the system. In order to obtain the maximum
value of the entropy function S’, the relation between the mean diameters Dq0 and the particle
diameter Dp is used as constraint[27]:


0
f n ( Dp ) Dpq dDp  Dqq0
(2)
According to the definition of fn(Dp), another constraint exists:


0
f n ( Dp )dDp  1
(3)
By solving equations. (1), (2) and (3), following is obtained:
f n ( Dp )  exp(a0  a1Dpq )
(4)
where a0 and a1 are Lagrange multipliers, which satisfy the following equations:
exp(a0 )  q
a1 
1 q
q
1
Dq 0 ( )
q
1
qDqq0
(5)
(6)
where Γ is the Gamma function. In order to compare the experimental data with the simulation data,
fv(Dp) is introduced as the volume based distribution. The relation between the fv(Dp) and the fn(Dp)
is
fv  Dp  
D 3p
D303
fn  Dp 
(7)
where D30 is the mean volumetric diameter. Therefore, the final expression for volume-based
distribution can be written based on the assumption that the Shannon’s entropy function is
maximized:
f v  Dp   q
q4
q
D3p
D (4 / q)
4
q0
exp(
Dq
qDqq0
)
(8)
From the Hinze-Kolmogorov theory, the Sauter mean diameter D32 is obtained:
D32 
D303
D202
(9)
where D20 is the mean surface diameter. Then the relation between the D32 and the Dq0 can be obtained
by combining equations (2) and (9):
D32  Dq 0  q q
(4 / q)
(3 / q)
(10)
Here, the parameter q is related to the width of the distribution and equals the distribution parameter
of the Rosin-Rammler distribution[27]. In order to estimate the value of q, the Rosin-Rammler
distribution is introduced:
1  Qi  exp(
di q
)
Y
(11)
where Qi is cumulative volume fraction for particle smaller than di. Y is a size parameter, which
measures the particle size below 63.2% of total volume[26],[37]. By plotting ln(1-Qi)-1 versus di/Y , the
value of q can be obtained.