Step 1: Recall the particle size classification of soils.
Soil particles are classified based on their size as follows:
Gravel: $>4.75 \, \text{mm} $
Sand: $ 0.075 \, \text{mm} \text{ to } 4.75 \, \text{mm} $
Silt: $ 0.002 \, \text{mm} \text{ to } 0.075 \, \text{mm} $
Clay: $<0.002 \, \text{mm} $
From this classification, the minimum size of silt particles is: $$ \boxed{0.002 \, \text{mm}} $$
Step 2: Evaluate other options.
$ 0.04 \, \text{mm} $: Lies within the silt range but not the minimum.
$ 0.075 \, \text{mm} $: Upper limit of silt; boundary with sand.
$ 0.005 \, \text{mm} $: Too large for clay, smaller than minimum silt.
Step 3: Select the correct option.
Based on standard soil particle size classification: $$ \boxed{0.002 \, \text{mm}} $$