eliminate space between tilde and letter in equations

The default "math type" of \sim is mathrel, i.e., that of a relational operator. (Another example of an operator of type mathrel is =.) The default math type of the letter A is mathord ("math ordinary"). As you've "discovered", TeX automatically inserts some whitespace -- in the amount of \thickmuskip -- between elements of types mathrel and mathord, respectively.

If you do not wish to get whitespace between \sim and A, simply write

{\sim}A

This setup changes the math type of \sim to mathord, and TeX does not insert extra whitespace between elements of type mathord.

Usually \sim is a relation like =. So the space is wanted:

\documentclass{article}

\usepackage{lmodern}
\usepackage{amsmath}

\begin{document}
\[ A = B \]
\[ A \sim B \]
\end{document}

But you could also define it as an ordinary symbol:

\documentclass{article}

\usepackage{lmodern}
\usepackage{amsmath}
\mathchardef\simsym"0218
\begin{document}
\[ A \sim B \]
\[ A \simsym B \]
\end{document}

For more information about math classes see, e.g., TeX by Topic section 38.3 and 23.3.

Instead of using \mathchardef you could also use

\newcommand*{\simsym}{\mathord\sim}

to define \simsym as the ordinary symbol variant of \sim:

\documentclass{article}

\usepackage{lmodern}
\usepackage{amsmath}
\newcommand*{\simsym}{\mathord\sim}
\begin{document}
\[ A \sim B \]
\[ A \simsym B \]
\end{document}

The result is exactly the same as above and would be the same like with

\documentclass{article}

\usepackage{lmodern}
\usepackage{amsmath}
\newcommand*{\simsym}{{\sim}}% note the extra {…}
\begin{document}
\[ A \sim B \]
\[ A \simsym B \]
\end{document}