The main problem is that seconds() has a resolution of 1 second so can't be used to do what you want. The following changes will do the trick.

Add three new variables above your setup()

1. long baseTime;
2. float currSecs, lastSecs;

Add the folowing lines as the last lines in setup()

1.   // Calculate the number of millis for the last whole minute
2.   // ie when second() would have returned 0
3.   int baseTime = millis() - second() *1000;
4.   // Initialise these so they work in draw()
5.   lastSecs = currSecs = ((millis() - baseTime)`000)/1000.0;

In setup remove the frameRate command as you will no longer need it.

The rest of the changes are in draw()

Add this line after the 2 text statements and before the commands to calculate the arcs

1. // Calculate the current number of seconds as a float 
   
2. currSecs = ((millis() - baseTime)`000)/1000.0;

Also in draw change the secondAngle calculation to

1.   float secondAngle = map(currSecs, 0, 60, 0, TWO_PI) - HALF_PI;

Finally at the end of draw where you test for seconds() == 0 change this to

1.   if(currSecs < lastSecs) {
2.     stroke(255);
3.     ellipse(arc_x, arc_y, secondDiameter, secondDiameter);
4.   }
5.   lastSecs = currSecs;

This will change the arc when the second roll over from 59.999 > 0.0 otherwise the second hand gets stuck for 1 second between 0>1 second. Finally remember the number of seconds in lastSecs so this test will work next time.

I tried it out and it works smoothly.

Nice sketch by the way.