The solar day fluctuates very slightly due to complications in the earth’s rotation. Most of these fluctuations affect both solar and sidereal days; I think the eccentricity of Earth’s orbit also affects solar time.
[For those who are unfamiliar with this, siderial time is measured by Earth’s rotation against the (almost) fixed stars, whereas solar time, which is measured by UT1, is measured by its rotation against the Sun. Because the Earth also orbits the Sun in the same direction as its rotation, it takes one additional revolution to complete a solar year than it does a siderial year. This causes the solar day to be about four minutes longer than a siderial day.]
The sidereal day is stretching out quite predictably, and does not include leap seconds. So in sidereal time, the longest day of the year is December 31. I think this is also true of solar time in years without leap seconds. Note that most years do now have a leap second (35 of them since 1967).
The precession of the equinoxes is the reason why the "first point of Aries" is now in Pisces, and why astrological signs are one constellation off from where the sun actually is in a given part of the year. It was first discovered by the ancient Greek astronomer Hipparchus--which is why you'll never hear Obama talk about it (can't credit Muslim scientists for the discovery).