Plotting in matplotlib

@AnttiA's answer is basically correct, but can be easily misunderstood, as can be seen from the OP's comment. Therefore here the complete code altered such that a plot is actually produced. Instead of making Z a list, define another variable as list, say Z_all = [], and then append the updated Z-values to that list. The same can be done for the time variable, i.e. time_all = np.arange(1,100,dt). Finally, take the plot command out of the loop and plot the entire data series at once.

Note that in your example you don't really have a series of random numbers, you pull one fixed number for one fixed seed, thus the plot is not really meaningful (it appears to be producing a straight line). Trying to interpret your intentions correctly, you probably want a series of random numbers that is as long as your time series. This is most easily done using np.random.normal

There are a lot of other ways that your code could be optimised. For instance, all mathematical functions from the math module are also found in numpy, so you could just not import math at all. The same goes for pandas. Also, you define some constant values inside the for-loop, which could be computed once before the loop. Lastly, @AnttiA is probably right, that you want time on the x axis and Z on the y axis. I therefore generate two plots -- on the left time over Z and on the right Z over time. Now finally the altered code:

import numpy as np
#from random import gauss
#from random import seed
#from pandas import Series
import matplotlib.pyplot as plt 
#import math

###variable declaration
R=0.000001 #radiaus of particle
beta=0.23 # shape factor
Ad=9.2#characteristic nanoscale defect area
gamma=4*10**-2 #surface tension
tau=.0001 #line tension

phi=-(np.pi/2)#spatial perturbation
lamda=((Ad)/(2*3.14*R)) #averge contcat line position

mu=0.001#viscosity of liquid
lamda_m=10**-9# characteristic size of adsorption site
KbT=(1.38**-24)*293 # boltzman constant with tempartaure
nu=0.001#moleculer volume in liquid phase 1
khi=3 #scaling factor
#deltaF=(beta*gamma*Ad)#surface energy perturbation
deltaF=19*KbT


##quantities moved out of the for-loop:
theta_e=((np.pi*110)/180) #equilibrium contact angle
Z_e=-R*np.cos(theta_e)#equilibrium position of particle
C=3.14*gamma*(R-Z_e) #additive constant
w_a=gamma*lamda_m**2*(1-np.cos(theta_e)) #work of adhesion

#########################################
Z=0.0000001 #particle position
##time=1
dt=1

Z_all = []
time_all = np.arange(1, 100, dt)
# seed random number generator
# seed(0)
np.random.seed(0)
# create white noise series
##series = [gauss(0.0, 1.0) for i in range(1)]
##series = Series(series)
series = np.random.normal(0.0, 1.0, len(time_all))

for time, S in zip(time_all,series):
#####simulation loop#######
    Z_all.append(Z)
    theta=np.arccos(-Z/R) #contact angle
    Fsz= (gamma*np.pi*(Z-Z_e)**2)+(tau*2*np.pi*np.sqrt(R**2-Z**2))+C
    Fz=Fsz+(0.5*deltaF*np.sin((2*np.pi/lamda)*(Z-Z_e)-phi))#surface force
    #dFz=(((gamma*Ad)/2)*np.sin(2*np.pi/lamda))+((Z-Z_e)*(2*gamma*np.pi))-((tau*2*np.pi*Z)/(np.sqrt(R**2-Z**2)))
    dFz=(deltaF*np.sin(2*np.pi/lamda))+((Z-Z_e)*(2*gamma*np.pi))-((tau*2*np.pi*Z)/(np.sqrt(R**2-Z**2)))
    epsilon_z=2*np.pi*R*np.sin(theta)*mu*(nu/(lamda_m**3))*np.exp(w_a/KbT)#transitional drag
    epsilon_s=khi*mu*((4*np.pi**2*R**2)/np.sqrt(Ad))*(1-(Z/R)**2)
    epsilon=epsilon_z+epsilon_s
    Ft=np.sqrt(2*KbT*epsilon)*S #series #thermal force
    v=(dFz+Ft)/epsilon ##new velocity
    Z=Z+v*dt #new position    
    print('z=',Z)
    print('v=',v)
    print('Fz=',Fz)
    print('dFz',dFz)
    print('time',time)    


fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(8,4))

axes[0].plot(Z_all,time_all)
axes[0].set_xlabel('Z')
axes[0].set_ylabel('t')

axes[1].plot(time_all, Z_all)
axes[1].set_xlabel('t')
axes[1].set_ylabel('Z')

fig.tight_layout()

plt.show()

The result looks like this: