Particle motion characteristic I'm making a particle motion raffling normal numbers. The normal random numbers raffled are the angles of the directions that the particle is going. The particle speed is constant. Look how this is done:
//initializing values  

initial_mean = 0 //initial direction
standard_deviation = pi
number_of_steps = 500
particle_speed = 50 // per second
positions_vector = new Positions_List
positions_vector.push([100, 100]) // push initial position


//starting the simulation

mean = initial_mean
foreach step in number_of_steps, do {
    angle = get_random_normal_number(standard_deviation, mean)
    angle = fix_angle(angle) // keep the angle between 0 and 2*pi
    x = cos(angle)*particle_speed + positions_vector.last_position.x
    y = sin(angle)*particle_speed + positions_vector.last_position.y
    mean = angle //update the direction
    positions_vector.push([x, y])
}

//plotting the result

plot(positions_vector)

Here is a new situation: suppose I need to improve the motion resolution. To do it, I need to calculate more points, but still making the particle move the same final distance. So I created a resolution factor that I multiply the number of steps by it and divide the particle speed by it too, so I can have more steps and moving the same final distance. Look:
mean = initial_mean
resolution_factor = 4 // If I want the motion 4 times more "precise"
foreach step in (number_of_steps * resolution_factor), do {
    angle = get_random_normal_number(standard_deviation, mean)
    angle = fix_angle(angle)
    x = cos(angle)*(particle_speed/resolution_factor) + positions_vector.last_position.x
    y = sin(angle)*(particle_speed/resolution_factor) + positions_vector.last_position.y
    mean = angle
    positions_vector.push([x, y])
}

Now, in this case, I'm gonna have 2000 steps, instead of 500. But the speed, now, is 1/4 lower to compensate the steps increase.
Here is the problem: when I increase this resolution factor, the motion "curvature" is "higher". Look these plots to show it happening:



QUESTION: Is there a way to keep the motion "characteristic" the same regardless of the resolution factor? Perhaps I need to scale the distribution reducing the standard deviation to get smaller angles based on the value of the resolution factor, but what would be the relation between the standard deviation and the resolution factor and why? Not only answers, but hints will also be of great help!
Working code in python

Here are some plots after the answer. Worked fine!
Resolution factor = 5, 20 and 100



 A: You need to scale the standard deviation so that the rate of curvature will remain the same. You are summing a bunch of normally distributed random variables, $X_i\sim N(0,\sigma^2)$, to obtain the angle $X$ after $n$ steps:
$$ X=\sum_{i=1}^n X_i $$
This will also be normally distributed with variance $n\sigma^2$. Now consider a larger sum:
$$ X=\sum_{i=1}^{tn} X_i $$
which will be normally distributed with variance $tn\sigma^2$.
You want $tn\sigma^2=n\sigma^2$, so set $\sigma\leftarrow\sigma/\sqrt{t}$.
A: The above is nice answer from statistical point-of-view. It works for small angles. Congrats! But you should keep in mind that this is a approximation which is convenient to be controlled. You are dealing with two kind of groups: rotation - translation - rotation - translation (RTRT). The case would be exact and the "dynamics" would be independent of the dice game if you were dealing with commutating groups 2D(1 axis) rotation - rotation - rotation - rotation (RRRR). In other words, the above sum of stochastic variables is not 'exact' since that sum is commutative and the applied operations (RT)(RT) are not for higher angles. So, it is convenient to control the deviations... I guess that, at the end of the day  the "distribution of positions" would be wider.
