A javascript function that simply runs a php function on the plugin

In your JS code, you need an action so it knows what function to execute.

var data = {
    'action': 'my_action',
    'nothing': 'nothing'
    'whatever': ajax_object.we_walue
  }

You should see how the action corresponds to the hook you used in add_action. But I would also add a second action:

add_action('wp_ajax_my_action', 'back_php');
add_action('wp_ajax_no_priv_my_action', 'back_php');

This will ensure that all users will be able to trigger the AJAX, not just those that are logged in.

If you don’t need the PHP function to use any data, you could simply pass an empty string. But to ensure it works, I would have the PHP function return something:

function back_php()
{
    echo "text from PHP";
    wp_die();
}

…and then catch it in the AJAX success function:

jQuery.post(ajaxurl, data, function(response) {
  alert(response);
}

Hope this helps.