special relativity - Is there an upper limit to temperature in thermodynamics or statistical mechanics

In many presentations of statistical mechanics where we have a system of particles having mass, such as the molecules of an ideal gas, the temperature is often equated to the average relative velocity of the particles. However no such velocity can exceed that of light. Is there therefore an absolute upper bound to temperature as well as an absolute zero?

Answer

Temperature is proportional to the average kinetic energy, not velocity, of the particles. Kinetic energy is unbounded; it goes to infinity as velocity approaches light-speed, proportional to $(1 - v^2/c^2)^{-1/2}$.

puzzle identification - What is this magic trick with 6 cards featuring numbers between 1 and 60?

I hope this is on topic. I have a set of kids' magic tricks but I have lost the instructions! I was hoping this might be a good place to get some help.

Here is the first one.

What is the trick and how does it work?

Answer

The magic trick is as follows.

Ask someone to think of a number. They should find all the cards with that number on, and give them to you. All you need do is add up the top left hand number on each of these cards. This will be the number they chose.

Source.

Explanation:

The first number on each card is a power of 2 (2,4,8,16,32). Every natural number up to 63 is a sum of some of these numbers (just write it in base 2). The rest of the numbers on each card are chosen precisely to make this work: on the card whose first number is $2^n$, the numbers appearing are exactly those whose binary expression has a $1$ in th $n$th position. How high the numbers go on each card is irrelevant, as long as the upper bound is the same on each one (here it seems to be 59).

Why probability of detection by performing unambiguous quantum measurement is less than random guess in mesoscopic quantum regime?

In mesoscopic quantum regime (mean photon number 10000) and non-orthogonal coherent state(number of non-orthogonal coherent state 2000), why probability of detection by performing quantum unambiguous measurement is less than random guess(1/2000). My question is that, if probability of detection is less than random guess following any particular quantum measurement process, then it will be better to follow random-guess(no-measurement).

Answer

The title of the paper is "Optimum Unambiguous Discrimination Between Linearly Independent Symmetric States".

Random guessing is an ambiguous discrimination procedure. It can return the wrong answer without telling you that it failed.

More generally, unambiguous discrimination procedures are less likely to succeed because removing any chance of accidental failure also sacrifices any chance of accidental success. You don't get the free $+1/n$ boost that ambiguous procedures get from falling back to random guessing when all else fails. They start from zero, and when the states are really close and have trace distance $<< 1/n$ that makes quite a big difference.

word - I wish, I yearn, for an answer to this riddle

In a creature is my natural home.
Have me in your meal and you'll become me.
(With me, this line would be adjacent.)
Experts take more time with me.
In winter wear, I might be half a mile.

Answer

I really hope the answer is

long

In a creature is my natural home.

belonging - meaning your natural home, the creature is a being.

Have me in your meal and you'll become me.

elongating - meaning becoming long, and eating as in your meal.

(With me, this line would be adjacent.)

alongside - meaning adjacent, the remark is an aside.

Experts take more time with me.

prolong/s - take more time, the experts are pro/s.

In winter wear, I might be half a mile.

Furlongs - furs are winter wear, four furlongs is half a mile.

Title: I wish, I yearn, for an answer to this riddle

long - to wish or yearn for something (such a puzzle from jafe or Rand).

cosmology - Why does the early universe look so big in a telescope?

As we look into the Deep Field, we see the Hubble photos of early galaxies up to only $400$ millions years old:

Wikipedia: List of the most distant astronomical objects

Because there is no center of the universe expansion, we can look for early galaxies anywhere in the sky. The earliest galaxies appear to us as the most distant in every direction. Therefore the early universe appears from our viewpoint as the most distant and thus the largest sphere with the radius, as we see it, of $13.4$ billion light-years (based on the most distant object in the above Wikipedia link).

Oversimplifying the spacetime geometry for a moment, the circumference of the early universe appears to us as $84$ billion light years ($2\pi\cdot 13.4 $) at the age of only $400$ million years. Clearly this calculation is incorrect. Obviously the universe was much smaller. However, we can see numerous galaxies along this circumference of our viewpoint. So either the distances we see between the early galaxies are much smaller than they appear to us in the sky or else the early universe looks way too big for its age.

What is the logical explanation of this illusion?

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EDIT

Since this question is misinterpreted and misguidedly closed, I have reposted it here:

Is the universe a giant telescope?

Lagrangians not of the form $T-U$

My Physics teacher was reluctant to define Lagrangian as Kinetic Energy minus Potential Energy because he said that there were cases where a system's Lagrangian did not take this form. Are you are aware of any such examples?

Update: Here I'm of course assuming that $T$ and $U$ stands for the kinetic and the potential energy, respectively. Also:

1. 
   

   adding a total time derivative term to the Lagrangian, or

   
   


2. 
   

   scaling the Lagrangian with a non-zero multiplicative constant

   
   

do not change the Euler-Lagrange equations, as Dilaton and dmckee points out in the comments. Needless to say that I'm not interested in such trivial modifications (1&2).

electrostatics - Capacitance of two non parallel plates

What is the formula for capacitance of two non parallel plates at an angle with each other?If the plates were parallel then the value can be calculated as (PermittivityX area of one plate)/distance between them.But what happens in case the plates are tilted at an angle?The question came to mind while trying to understand electrostatic separator.What would be the derivation of the formula for capacitance of two non parallel plates placed at an angle?

I did get a method from http://www.davidpublishing.com/davidpublishing/upfile/12/15/2011/2011121573197833.pdf Equation 6 from the above link above helps but it is independent of the length of the plates which doesn't seem likely.